AI Data Privacy Concerns — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.
Face Swap Live is a mobile app created by Laan Labs that enables users to swap faces with another person in real-time using the device's camera. It was released on December 14, 2015. In addition to swapping faces with another person, the app enables users to create videos using a set of bundled live filters. The app is available on iOS and Android devices. Face Swap Live was named Apple's #2 best-selling paid app in 2016.
Read more →
In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep network can model the complex, non-linear relationships often found in real-world data. The best-known version of the theorem applies to feedforward networks with a single hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU), then the network can act as a "universal approximator." Universality is achieved by increasing the number of neurons in the hidden layer, making the network "wider." Other versions of the theorem show that universality can also be achieved by keeping the network's width fixed but increasing its number of layers, making it "deeper." These are existence theorems. They guarantee that a network with the right structure exists, but they do not provide a method for finding the network's parameters (training it), nor do they specify exactly how large the network must be for a given function. Finding a suitable network remains a practical challenge that is typically addressed with optimization algorithms like backpropagation. == Setup == Artificial neural networks are combinations of multiple simple mathematical functions that implement more complicated functions from (typically) real-valued vectors to real-valued vectors. The spaces of multivariate functions that can be implemented by a network are determined by the structure of the network, the set of simple functions, and its multiplicative parameters. A great deal of theoretical work has gone into characterizing these function spaces. Most universal approximation theorems are in one of two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("bounded depth and bounded width" case). == History == === Arbitrary width === The first results concerned the arbitrary width case. Ken-ichi Funahashi (May 1989) showed that Rumelhart–Hinton–Williams type backpropagation networks possess universal approximation capability with a class of sigmoidal activation functions, extending the result to multi-output mappings as well. Kurt Hornik, Maxwell Stinchcombe, and Halbert White (July 1989) showed that multilayer feed-forward networks with as few as one hidden layer are universal approximators, provided that the activation function satisfies certain conditions. George Cybenko (December 1989) independently established a related result for sigmoid activation functions using functional-analytic methods. Hornik also showed in 1991 that it is not the specific choice of the activation function but rather the multilayer feed-forward architecture itself that gives neural networks the potential of being universal approximators. Moshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function. === Arbitrary depth === The arbitrary depth case was also studied by a number of authors such as Gustaf Gripenberg in 2003, Dmitry Yarotsky, Zhou Lu et al in 2017, Boris Hanin and Mark Sellke in 2018 who focused on neural networks with ReLU activation function. In 2020, Patrick Kidger and Terry Lyons extended those results to neural networks with general activation functions such, e.g. tanh or GeLU. One special case of arbitrary depth is that each composition component comes from a finite set of mappings. In 2024, Cai constructed a finite set of mappings, named a vocabulary, such that any continuous function can be approximated by compositing a sequence from the vocabulary. This is similar to the concept of compositionality in linguistics, which is the idea that a finite vocabulary of basic elements can be combined via grammar to express an infinite range of meanings. === Bounded depth and bounded width === The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999. They showed that there exists an analytic sigmoidal activation function such that two hidden layer neural networks with bounded number of units in hidden layers are universal approximators. In 2018, Guliyev and Ismailov constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with fewer units in hidden layers. In 2018, they also constructed single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this does not apply for multivariable functions. In 2022, Shen et al. obtained precise quantitative information on the depth and width required to approximate a target function by deep and wide ReLU neural networks. === Quantitative bounds === The question of minimal possible width for universality was first studied in 2021, Park et al obtained the minimum width required for the universal approximation of Lp functions using feed-forward neural networks with ReLU as activation functions. Similar results that can be directly applied to residual neural networks were also obtained in the same year by Paulo Tabuada and Bahman Gharesifard using control-theoretic arguments. In 2023, Cai obtained the optimal minimum width bound for the universal approximation. For the arbitrary depth case, Leonie Papon and Anastasis Kratsios derived explicit depth estimates depending on the regularity of the target function and of the activation function. === Kolmogorov network === The Kolmogorov–Arnold representation theorem is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. Robert Hecht-Nielsen showed that a three-layer neural network can approximate any continuous multivariate function. This was extended to the discontinuous case by Vugar Ismailov. In 2024, Ziming Liu and co-authors showed a practical application. === Reservoir computing and quantum reservoir computing === In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons and spiking neurons, respectively. In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. === Variants === Variants include discontinuous activation functions, noncompact domains, certifiable networks, random neural networks, and alternative network architectures and topologies. The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. For input dimension d x {\displaystyle d_{x}} and output dimension d y {\displaystyle d_{y}} the minimum width required for the universal approximation of the Lp functions is exactly m a x { d x + 1 , d y } {\displaystyle max\{d_{x}+1,d_{y}\}} (for a ReLU network). More generally this also holds if both ReLU and a threshold activation function are used. Universal function approximation on graphs (or rather on graph isomorphism classes) by popular graph convolutional neural networks (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation with certain injective properties is sufficient for universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying O ( | V | ⋅ | E | ) {\displaystyle {\mathcal {O}}(\left|V\right|\cdot \left|E\right|)} -runtime method that performed at state of the art on a collection of benchmarks (where V {\displaystyle V} and E {\displaystyle E} are the sets of nodes and edges of the graph respectively). There are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis functions, or neural networks with specific properties. == Arbitrary-width case == A universal approximation theorem formally states that a family of neural network funct
Read more →
A self-organizing map (SOM) or self-organizing feature map (SOFM) is an unsupervised machine learning technique used to produce a low-dimensional (typically two-dimensional) representation of a higher-dimensional data set while preserving the topological structure of the data. For example, a data set with p {\displaystyle p} variables measured in n {\displaystyle n} observations could be represented as clusters of observations with similar values for the variables. These clusters then could be visualized as a two-dimensional "map" such that observations in proximal clusters have more similar values than observations in distal clusters. This can make high-dimensional data easier to visualize and analyze. A SOM is a type of artificial neural network but is trained using competitive learning rather than the error-correction learning (e.g., backpropagation with gradient descent) used by other artificial neural networks. The SOM was introduced by the Finnish professor Teuvo Kohonen in the 1980s and therefore is sometimes called a Kohonen map or Kohonen network. The Kohonen map or network is a computationally convenient abstraction building on biological models of neural systems from the 1970s and morphogenesis models dating back to Alan Turing in the 1950s. SOMs create internal representations reminiscent of the cortical homunculus, a distorted representation of the human body, based on a neurological "map" of the areas and proportions of the human brain dedicated to processing sensory functions, for different parts of the body. == Overview == Self-organizing maps, like most artificial neural networks, operate in two modes: training and mapping. First, training uses an input data set (the "input space") to generate a lower-dimensional representation of the input data (the "map space"). Second, mapping classifies additional input data using the generated map. The goal of training is to represent an input space with p dimensions as a map space with n dimensions, where p > n. Specifically, an input space with p variables is said to have p dimensions. A map space consists of components called "nodes" or "neurons", which are arranged as a hexagonal or rectangular grid with two dimensions. The number of nodes and their arrangement are specified beforehand based on the larger goals of the analysis and exploration of the data. Each node in the map space is associated with a "weight" vector, which is the position of the node in the input space. While nodes in the map space stay fixed, training consists in moving weight vectors toward the input data (reducing a distance metric such as Euclidean distance) without spoiling the topology induced from the map space. After training, the map can be used to classify additional observations for the input space by finding the node with the closest weight vector (smallest distance metric) to the input space vector. == Learning algorithm == The goal of learning in the self-organizing map is to cause different parts of the network to respond similarly to certain input patterns. This is partly motivated by how visual, auditory or other sensory information is handled in separate parts of the cerebral cortex in the human brain. The weights of the neurons are initialized either to small random values or sampled evenly from the subspace spanned by the two largest principal component eigenvectors. With the latter alternative, learning is much faster because the initial weights already give a good approximation of SOM weights. The network must be fed a large number of example vectors that represent, as close as possible, the kinds of vectors expected during mapping. The examples are usually administered several times as iterations. The training utilizes competitive learning. When a training example is fed to the network, its Euclidean distance to all weight vectors is computed. The neuron whose weight vector is most similar to the input is called the best matching unit (BMU). The weights of the BMU and neurons close to it in the SOM grid are adjusted towards the input vector. The magnitude of the change decreases with time and with the grid-distance from the BMU. The update formula for a neuron v with weight vector Wv(s) is W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} , where s is the step index, t is an index into the training sample, u is the index of the BMU for the input vector D(t), α(s) is a monotonically decreasing learning coefficient; θ(u, v, s) is the neighborhood function which gives the distance between the neuron u and the neuron v in step s. Depending on the implementations, t can scan the training data set systematically (t is 0, 1, 2...T-1, then repeat, T being the training sample's size), be randomly drawn from the data set (bootstrap sampling), or implement some other sampling method (such as jackknifing). The neighborhood function θ(u, v, s) (also called function of lateral interaction) depends on the grid-distance between the BMU (neuron u) and neuron v. In the simplest form, it is 1 for all neurons close enough to BMU and 0 for others, but the Gaussian and Mexican-hat functions are common choices, too. Regardless of the functional form, the neighborhood function shrinks with time. At the beginning when the neighborhood is broad, the self-organizing takes place on the global scale. When the neighborhood has shrunk to just a couple of neurons, the weights are converging to local estimates. In some implementations, the learning coefficient α and the neighborhood function θ decrease steadily with increasing s, in others (in particular those where t scans the training data set) they decrease in step-wise fashion, once every T steps. This process is repeated for each input vector for a (usually large) number of cycles λ. The network winds up associating output nodes with groups or patterns in the input data set. If these patterns can be named, the names can be attached to the associated nodes in the trained net. During mapping, there will be one single winning neuron: the neuron whose weight vector lies closest to the input vector. This can be simply determined by calculating the Euclidean distance between input vector and weight vector. While representing input data as vectors has been emphasized in this article, any kind of object which can be represented digitally, which has an appropriate distance measure associated with it, and in which the necessary operations for training are possible can be used to construct a self-organizing map. This includes matrices, continuous functions or even other self-organizing maps. === Algorithm === Randomize the node weight vectors in a map For s = 0 , 1 , 2 , . . . , λ {\displaystyle s=0,1,2,...,\lambda } Randomly pick an input vector D ( t ) {\displaystyle {D}(t)} Find the node in the map closest to the input vector. This node is the best matching unit (BMU). Denote it by u {\displaystyle u} For each node v {\displaystyle v} , update its vector by pulling it closer to the input vector: W v ( s + 1 ) = W v ( s ) + θ ( u , v , s ) ⋅ α ( s ) ⋅ ( D ( t ) − W v ( s ) ) {\displaystyle W_{v}(s+1)=W_{v}(s)+\theta (u,v,s)\cdot \alpha (s)\cdot (D(t)-W_{v}(s))} The variable names mean the following, with vectors in bold, s {\displaystyle s} is the current iteration λ {\displaystyle \lambda } is the iteration limit t {\displaystyle t} is the index of the target input data vector in the input data set D {\displaystyle \mathbf {D} } D ( t ) {\displaystyle {D}(t)} is a target input data vector v {\displaystyle v} is the index of the node in the map W v {\displaystyle \mathbf {W} _{v}} is the current weight vector of node v {\displaystyle v} u {\displaystyle u} is the index of the best matching unit (BMU) in the map θ ( u , v , s ) {\displaystyle \theta (u,v,s)} is the neighbourhood function, α ( s ) {\displaystyle \alpha (s)} is the learning rate schedule. The key design choices are the shape of the SOM, the neighbourhood function, and the learning rate schedule. The idea of the neighborhood function is to make it such that the BMU is updated the most, its immediate neighbors are updated a little less, and so on. The idea of the learning rate schedule is to make it so that the map updates are large at the start, and gradually stop updating. For example, if we want to learn a SOM using a square grid, we can index it using ( i , j ) {\displaystyle (i,j)} where both i , j ∈ 1 : N {\displaystyle i,j\in 1:N} . The neighborhood function can make it so that the BMU updates in full, the nearest neighbors update in half, and their neighbors update in half again, etc. θ ( ( i , j ) , ( i ′ , j ′ ) , s ) = 1 2 | i − i ′ | + | j − j ′ | = { 1 if i = i ′ , j = j ′ 1 / 2 if | i − i ′ | + | j − j ′ | = 1 1 / 4 if | i − i ′ | + | j − j ′ | = 2 ⋯ ⋯ {\displaystyle \theta ((i,j),(i',j'),s)={\frac {1}{2^{|i-i'|+|j-j'|}}}={\begin{cases}1&{\text{if }}i=i',j=j'\\1/2&{\text{if
Read more →
An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-
Read more →
80 Million Tiny Images is a dataset intended for training machine-learning systems constructed by Antonio Torralba, Rob Fergus, and William T. Freeman in a collaboration between MIT and New York University. It was published in 2008. The dataset has size 760 GB. It contains 79,302,017 32×32-pixel color images, scaled down from images scraped from the World Wide Web over 8 months. The images are classified into 75,062 classes. Each class is a non-abstract noun in WordNet. Images may appear in more than one class. The dataset was motivated by non-parametric models of neural activations in the visual cortex upon seeing images. The CIFAR-10 dataset uses a subset of the images in this dataset, but with independently generated labels, as the original labels were not reliable. The CIFAR-10 set has 6000 examples of each of 10 classes, and the CIFAR-100 set has 600 examples of each of 100 non-overlapping classes. == Construction == It was first reported in a technical report in April 2007, during the middle of the construction process, when there were only 73 million images. The full dataset was published in 2008. They began with all 75,846 non-abstract nouns in WordNet, and then for each of these nouns, they scraped 7 image search engines: Altavista, Ask.com, Flickr, Cydral, Google, Picsearch, and Webshots. After 8 months of scraping, they obtained 97,245,098 images. Since they did not have enough storage, they downsized the images to 32×32 as they were scraped. After gathering, they removed images with zero variance and intra-word duplicate images, resulting in the final dataset. Out of the 75,846 nouns, only 75,062 classes had any results, so the other nouns did not appear in the final dataset. The number of images per noun follows a Zipf-like distribution, with 1056 images per noun on average. To prevent a few nouns taking up too many images, they put an upper bound of at most 3000 images per noun. == Retirement == The 80 Million Tiny Images dataset was retired from use by its creators in 2020, after a paper by researchers Abeba Birhane and Vinay Prabhu found that some of the labeling of several publicly available image datasets, including 80 Million Tiny Images, contained racist and misogynistic slurs which were causing models trained on them to exhibit racial and sexual bias. The dataset also contained offensive images. Following the release of the paper, the dataset's creators removed the dataset from distribution, and requested that other researchers not use it for further research and to delete their copies of the dataset.
Read more →
Aleph (A Learning Engine for Proposing Hypotheses) is an inductive logic programming system introduced by Ashwin Srinivasan in 2001. As of 2022 it is still one of the most widely used inductive logic programming systems. It is based on the earlier system Progol. == Learning task == The input to Aleph is background knowledge, specified as a logic program, a language bias in the form of mode declarations, as well as positive and negative examples specified as ground facts. As output it returns a logic program which, together with the background knowledge, entails all of the positive examples and none of the negative examples. == Basic algorithm == Starting with an empty hypothesis, Aleph proceeds as follows: It chooses a positive example to generalise; if none are left, it aborts and outputs the current hypothesis. Then it constructs the bottom clause, that is, the most specific clause that is allowed by the mode declarations and covers the example. It then searches for a generalisation of the bottom clause that scores better on the chosen metric. It then adds the new clause to the hypothesis program and removes all examples that are covered by the new clause. == Search algorithm == Aleph searches for clauses in a top-down manner, using the bottom clause constructed in the preceding step to bound the search from below. It searches the refinement graph in a breadth-first manner, with tunable parameters to bound the maximal clause size and proof depth. It scores each clause using one of 13 different evaluation metrics, as chosen in advance by the user.
Read more →
In machine learning, the polynomial kernel is a kernel function commonly used with support vector machines (SVMs) and other kernelized models, that represents the similarity of vectors (training samples) in a feature space over polynomials of the original variables, allowing learning of non-linear models. Intuitively, the polynomial kernel looks not only at the given features of input samples to determine their similarity, but also combinations of these. In the context of regression analysis, such combinations are known as interaction features. The (implicit) feature space of a polynomial kernel is equivalent to that of polynomial regression, but without the combinatorial blowup in the number of parameters to be learned. When the input features are binary-valued (booleans), then the features correspond to logical conjunctions of input features. == Definition == For degree-d polynomials, the polynomial kernel is defined as K ( x , y ) = ( x T y + c ) d {\displaystyle K(\mathbf {x} ,\mathbf {y} )=(\mathbf {x} ^{\mathsf {T}}\mathbf {y} +c)^{d}} where x and y are vectors of size n in the input space, i.e. vectors of features computed from training or test samples and c ≥ 0 is a free parameter trading off the influence of higher-order versus lower-order terms in the polynomial. When c = 0, the kernel is called homogeneous. (A further generalized polykernel divides xTy by a user-specified scalar parameter a.) As a kernel, K corresponds to an inner product in a feature space based on some mapping φ: K ( x , y ) = ⟨ φ ( x ) , φ ( y ) ⟩ {\displaystyle K(\mathbf {x} ,\mathbf {y} )=\langle \varphi (\mathbf {x} ),\varphi (\mathbf {y} )\rangle } The nature of φ can be seen from an example. Let d = 2, so we get the special case of the quadratic kernel. After using the multinomial theorem (twice—the outermost application is the binomial theorem) and regrouping, K ( x , y ) = ( ∑ i = 1 n x i y i + c ) 2 = ∑ i = 1 n ( x i 2 ) ( y i 2 ) + ∑ i = 2 n ∑ j = 1 i − 1 ( 2 x i x j ) ( 2 y i y j ) + ∑ i = 1 n ( 2 c x i ) ( 2 c y i ) + c 2 {\displaystyle K(\mathbf {x} ,\mathbf {y} )=\left(\sum _{i=1}^{n}x_{i}y_{i}+c\right)^{2}=\sum _{i=1}^{n}\left(x_{i}^{2}\right)\left(y_{i}^{2}\right)+\sum _{i=2}^{n}\sum _{j=1}^{i-1}\left({\sqrt {2}}x_{i}x_{j}\right)\left({\sqrt {2}}y_{i}y_{j}\right)+\sum _{i=1}^{n}\left({\sqrt {2c}}x_{i}\right)\left({\sqrt {2c}}y_{i}\right)+c^{2}} From this it follows that the feature map is given by: φ ( x ) = ( x n 2 , … , x 1 2 , 2 x n x n − 1 , … , 2 x n x 1 , 2 x n − 1 x n − 2 , … , 2 x n − 1 x 1 , … , 2 x 2 x 1 , 2 c x n , … , 2 c x 1 , c ) {\displaystyle \varphi (x)=\left(x_{n}^{2},\ldots ,x_{1}^{2},{\sqrt {2}}x_{n}x_{n-1},\ldots ,{\sqrt {2}}x_{n}x_{1},{\sqrt {2}}x_{n-1}x_{n-2},\ldots ,{\sqrt {2}}x_{n-1}x_{1},\ldots ,{\sqrt {2}}x_{2}x_{1},{\sqrt {2c}}x_{n},\ldots ,{\sqrt {2c}}x_{1},c\right)} generalizing for ( x T y + c ) d {\displaystyle \left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}} , where x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} , y ∈ R n {\displaystyle \mathbf {y} \in \mathbb {R} ^{n}} and applying the multinomial theorem: ( x T y + c ) d = ∑ j 1 + j 2 + ⋯ + j n + 1 = d d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 d ! j 1 ! ⋯ j n ! j n + 1 ! y 1 j 1 ⋯ y n j n c j n + 1 = φ ( x ) T φ ( y ) {\displaystyle {\begin{alignedat}{2}\left(\mathbf {x} ^{T}\mathbf {y} +c\right)^{d}&=\sum _{j_{1}+j_{2}+\dots +j_{n+1}=d}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}{\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}y_{1}^{j_{1}}\cdots y_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\\&=\varphi (\mathbf {x} )^{T}\varphi (\mathbf {y} )\end{alignedat}}} The last summation has l d = ( n + d d ) {\displaystyle l_{d}={\tbinom {n+d}{d}}} elements, so that: φ ( x ) = ( a 1 , … , a l , … , a l d ) {\displaystyle \varphi (\mathbf {x} )=\left(a_{1},\dots ,a_{l},\dots ,a_{l_{d}}\right)} where l = ( j 1 , j 2 , . . . , j n , j n + 1 ) {\displaystyle l=(j_{1},j_{2},...,j_{n},j_{n+1})} and a l = d ! j 1 ! ⋯ j n ! j n + 1 ! x 1 j 1 ⋯ x n j n c j n + 1 | j 1 + j 2 + ⋯ + j n + j n + 1 = d {\displaystyle a_{l}={\frac {\sqrt {d!}}{\sqrt {j_{1}!\cdots j_{n}!j_{n+1}!}}}x_{1}^{j_{1}}\cdots x_{n}^{j_{n}}{\sqrt {c}}^{j_{n+1}}\quad |\quad j_{1}+j_{2}+\dots +j_{n}+j_{n+1}=d} == Practical use == Although the RBF kernel is more popular in SVM classification than the polynomial kernel, the latter is quite popular in natural language processing (NLP). The most common degree is d = 2 (quadratic), since larger degrees tend to overfit on NLP problems. Various ways of computing the polynomial kernel (both exact and approximate) have been devised as alternatives to the usual non-linear SVM training algorithms, including: full expansion of the kernel prior to training/testing with a linear SVM, i.e. full computation of the mapping φ as in polynomial regression; basket mining (using a variant of the apriori algorithm) for the most commonly occurring feature conjunctions in a training set to produce an approximate expansion; inverted indexing of support vectors. One problem with the polynomial kernel is that it may suffer from numerical instability: when xTy + c < 1, K(x, y) = (xTy + c)d tends to zero with increasing d, whereas when xTy + c > 1, K(x, y) tends to infinity.
Read more →
In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment. == Network architecture == Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} . The output of the network is then a scalar function of the input vector, φ : R n → R {\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} } , and is given by φ ( x ) = ∑ i = 1 N a i ρ ( | | x − c i | | ) {\displaystyle \varphi (\mathbf {x} )=\sum _{i=1}^{N}a_{i}\rho (||\mathbf {x} -\mathbf {c} _{i}||)} where N {\displaystyle N} is the number of neurons in the hidden layer, c i {\displaystyle \mathbf {c} _{i}} is the center vector for neuron i {\displaystyle i} , and a i {\displaystyle a_{i}} is the weight of neuron i {\displaystyle i} in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition) and the radial basis function is commonly taken to be Gaussian ρ ( ‖ x − c i ‖ ) = exp [ − β i ‖ x − c i ‖ 2 ] {\displaystyle \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta _{i}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]} . The Gaussian basis functions are local to the center vector in the sense that lim | | x | | → ∞ ρ ( ‖ x − c i ‖ ) = 0 {\displaystyle \lim _{||x||\to \infty }\rho (\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert )=0} i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron. Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of R n {\displaystyle \mathbb {R} ^{n}} . This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision. The parameters a i {\displaystyle a_{i}} , c i {\displaystyle \mathbf {c} _{i}} , and β i {\displaystyle \beta _{i}} are determined in a manner that optimizes the fit between φ {\displaystyle \varphi } and the data. === Normalization === ==== Normalized architecture ==== In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is φ ( x ) = d e f ∑ i = 1 N a i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N a i u ( ‖ x − c i ‖ ) {\displaystyle \varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}a_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} where u ( ‖ x − c i ‖ ) = d e f ρ ( ‖ x − c i ‖ ) ∑ j = 1 N ρ ( ‖ x − c j ‖ ) {\displaystyle u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{j=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{j}\right\Vert {\big )}}}} is known as a normalized radial basis function. ==== Theoretical motivation for normalization ==== There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density P ( x ∧ y ) = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) σ ( | y − e i | ) {\displaystyle P\left(\mathbf {x} \land y\right)={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,\sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}} where the weights c i {\displaystyle \mathbf {c} _{i}} and e i {\displaystyle e_{i}} are exemplars from the data and we require the kernels to be normalized ∫ ρ ( ‖ x − c i ‖ ) d n x = 1 {\displaystyle \int \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,d^{n}\mathbf {x} =1} and ∫ σ ( | y − e i | ) d y = 1 {\displaystyle \int \sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}\,dy=1} . The probability densities in the input and output spaces are P ( x ) = ∫ P ( x ∧ y ) d y = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) {\displaystyle P\left(\mathbf {x} \right)=\int P\left(\mathbf {x} \land y\right)\,dy={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and The expectation of y given an input x {\displaystyle \mathbf {x} } is φ ( x ) = d e f E ( y ∣ x ) = ∫ y P ( y ∣ x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)\ {\stackrel {\mathrm {def} }{=}}\ E\left(y\mid \mathbf {x} \right)=\int y\,P\left(y\mid \mathbf {x} \right)dy} where P ( y ∣ x ) {\displaystyle P\left(y\mid \mathbf {x} \right)} is the conditional probability of y given x {\displaystyle \mathbf {x} } . The conditional probability is related to the joint probability through Bayes' theorem P ( y ∣ x ) = P ( x ∧ y ) P ( x ) {\displaystyle P\left(y\mid \mathbf {x} \right)={\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}} which yields φ ( x ) = ∫ y P ( x ∧ y ) P ( x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)=\int y\,{\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}\,dy} . This becomes φ ( x ) = ∑ i = 1 N e i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N e i u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)={\frac {\sum _{i=1}^{N}e_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}e_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} when the integrations are performed. === Local linear models === It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order, φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) ρ ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} in the unnormalized and normalized cases, respectively. Here b i {\displaystyle \mathbf {b} _{i}} are weights to be determined. Higher order linear terms are also possible. This result can be written φ ( x ) = ∑ i = 1 2 N ∑ j = 1 n e i j v i j ( x − c i ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{2N}\sum _{j=1}^{n}e_{ij}v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}} where e i j = { a i , if i ∈ [ 1 , N ] b i j , if i ∈ [ N + 1 , 2 N ] {\displaystyle e_{ij}={\begin{cases}a_{i},&{\mbox{if }}i\in [1,N]\\b_{ij},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} and v i j ( x − c i ) = d e f { δ i j ρ ( ‖ x − c i ‖ ) , if i ∈ [ 1 , N ] ( x i j − c i j ) ρ ( ‖ x − c i ‖ ) , if i ∈ [ N + 1 , 2 N ] {\displaystyle v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}\delta _{ij}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [1,N]\\\left(x_{ij}-c_{ij}\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} in the unnormalized case and in the normalized case. Here δ i j {\displaystyle \delta _{ij}} is a Kronecker delta function defined as δ i j = { 1 , if i = j 0 , if i ≠ j {\displaystyle \delta _{ij}={\begin{cases}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{cases}}} . == Training == RBF networks are typically trained from pairs of input and target values x ( t ) , y ( t ) {\displaystyle \mathbf {x} (t),y(t)} , t = 1 , … , T {\displaystyle t=1,\dots ,T} by a two-step algorithm. In the first step, the center vectors c i {\displaystyle \mathbf {c} _{i}} of the RBF functions in the hidden layer
Read more →
WriterDuet is a screenwriting software for writing and editing screenplays and other forms of mass media. == History == WriterDuet was founded in 2013 by Guy Goldstein. In April 2015, WriterDuet acquired the domain for Scripped.com after they closed, citing a serious technical failure. In August 2016, WriterDuet released a localized version of its software in China. In May 2018, WriterDuet included Bechdel test analysis functions to address issues of gender diversity in the screenwriting industry. In 2018, WriterDuet published WriterSolo, an offline version of their app that runs on the browser and opens/saves files on the computer, Dropbox, Google Drive, and iCloud. In July 2019, WriterDuet made the WriterSolo browser app and desktop app available as pay-what-you-want under the web address FreeScreenwriting.com. == Features == WriterDuet is primarily used to outline, write, and format screenplays to the standards recommended by the AMPAS. It also supports formats for theater, novels, and video games. The software is powered by Firebase allowing users to write together in real-time from multiple devices. WriterDuet's main competitors in the screenwriting industry are Final Draft, Celtx, and Movie Magic Screenwriter.
Read more →
Constructing skill trees (CST) is a hierarchical reinforcement learning algorithm which can build skill trees from a set of sample solution trajectories obtained from demonstration. CST uses an incremental MAP (maximum a posteriori) change point detection algorithm to segment each demonstration trajectory into skills and integrate the results into a skill tree. CST was introduced by George Konidaris, Scott Kuindersma, Andrew Barto and Roderic Grupen in 2010. == Algorithm == CST consists of mainly three parts;change point detection, alignment and merging. The main focus of CST is online change-point detection. The change-point detection algorithm is used to segment data into skills and uses the sum of discounted reward R t {\displaystyle R_{t}} as the target regression variable. Each skill is assigned an appropriate abstraction. A particle filter is used to control the computational complexity of CST. The change point detection algorithm is implemented as follows. The data for times t ∈ T {\displaystyle t\in T} and models Q with prior p ( q ∈ Q ) {\displaystyle p(q\in Q)} are given. The algorithm is assumed to be able to fit a segment from time j + 1 {\displaystyle j+1} to t using model q with the fit probability P ( j , t , q ) {\displaystyle P(j,t,q)_{}^{}} . A linear regression model with Gaussian noise is used to compute P ( j , t , q ) {\displaystyle P(j,t,q)} . The Gaussian noise prior has mean zero, and variance which follows I n v e r s e G a m m a ( v 2 , u 2 ) {\displaystyle \mathrm {InverseGamma} \left({\frac {v}{2}},{\frac {u}{2}}\right)} . The prior for each weight follows N o r m a l ( 0 , σ 2 δ ) {\displaystyle \mathrm {Normal} (0,\sigma ^{2}\delta )} . The fit probability P ( j , t , q ) {\displaystyle P(j,t,q)} is computed by the following equation. P ( j , t , q ) = π − n 2 δ m | ( A + D ) − 1 | 1 2 u v 2 ( y + u ) u + v 2 Γ ( n + v 2 ) Γ ( v 2 ) {\displaystyle P(j,t,q)={\frac {\pi ^{-{\frac {n}{2}}}}{\delta ^{m}}}\left|(A+D)^{-1}\right|^{\frac {1}{2}}{\frac {u^{\frac {v}{2}}}{(y+u)^{\frac {u+v}{2}}}}{\frac {\Gamma ({\frac {n+v}{2}})}{\Gamma ({\frac {v}{2}})}}} Then, CST compute the probability of the changepoint at time j with model q, P t ( j , q ) {\displaystyle P_{t}(j,q)} and P j MAP {\displaystyle P_{j}^{\text{MAP}}} using a Viterbi algorithm. P t ( j , q ) = ( 1 − G ( t − j − 1 ) ) P ( j , t , q ) p ( q ) P j MAP {\displaystyle P_{t}(j,q)=(1-G(t-j-1))P(j,t,q)p(q)P_{j}^{\text{MAP}}} P j MAP = max i , q P j ( i , q ) g ( j − i ) 1 − G ( j − i − 1 ) , ∀ j < t {\displaystyle P_{j}^{\text{MAP}}=\max _{i,q}{\frac {P_{j}(i,q)g(j-i)}{1-G(j-i-1)}},\forall j
Deterministic blockmodeling is an approach in blockmodeling that does not assume a probabilistic model, and instead relies on the exact or approximate algorithms, which are used to find blockmodel(s). This approach typically minimizes some inconsistency that can occur with the ideal block structure. Such analysis is focused on clustering (grouping) of the network (or adjacency matrix) that is obtained with minimizing an objective function, which measures discrepancy from the ideal block structure. However, some indirect approaches (or methods between direct and indirect approaches, such as CONCOR) do not explicitly minimize inconsistencies or optimize some criterion function. This approach was popularized in the 1970s, due to the presence of two computer packages (CONCOR and STRUCTURE) that were used to "find a permutation of the rows and columns in the adjacency matrix leading to an approximate block structure". The opposite approach to deterministic blockmodeling is a stochastic blockmodeling approach.
Read more →
The multi-surface method (MSM) is a form of decision making using the concept of piecewise-linear separability of datasets to categorize data. == Introduction == Two datasets are linearly separable if their convex hulls do not intersect. The method may be formulated as a feedforward neural network with weights that are trained via linear programming. Comparisons between neural networks trained with the MSM versus backpropagation show MSM is better able to classify data. The decision problem associated linear program for the MSM is NP-complete. == Mathematical formulation == Given two finite disjoint point sets A , B ∈ R n {\displaystyle {\mathcal {A,B}}\in \mathbb {R} ^{n}} , find a discriminant, f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } such that f ( A ) > 0 , f ( B ) ≤ 0 {\displaystyle f({\mathcal {A}})>0,f({\mathcal {B}})\leq 0} . If the intersection of convex hulls of the two sets is the empty set, then it is possible to use a single linear program to obtain a linear discriminant of the form, f ( x ) = c x + γ {\displaystyle f(x)=cx+\gamma } . Usually, in real applications, the sets' convex hulls do intersect, and a (often non-convex) piecewise-linear discriminant can be used, through the use of several linear programs.
Read more →
Lexalytics, Inc. provides sentiment and intent analysis to an array of companies using SaaS and cloud based technology. Salience 6, the engine behind Lexalytics, was built as an on-premises, multi-lingual text analysis engine. It is leased to other companies who use it to power filtering and reputation management programs. In July, 2015 Lexalytics acquired Semantria to be used as a cloud option for its technology. In September, 2021 Lexalytics was acquired by CX company InMoment. == History == Lexalytics spun into existence in January 2003 out of a content management startup called Lightspeed. Lightspeed consolidated on America's West Coast. Jeff Catlin, a Lightspeed General Manager, and Mike Marshall, a Lighstpeed Principal Engineer, convinced investors to give them the East Coast company so as to avoid shutdown costs. Catlin and Marshall renamed the operation Lexalytics. Catlin took on the role of chief executive officer with Marshall working as Chief Technology Officer. Lexalytics opted to not accept venture cash. Instead, the company initially shared sales and marketing expenses with U.K. based document management company Infonic. The partner companies soon formed a joint venture in July 2008, which was later dissolved. Since then, Lexalytics has worked with many other companies, like Bottlenose, Salesforce, Thomson Reuters, Oracle and DataSift. Relationships with social media monitoring companies like Datasift tend to find Lexalytics’ Salience engine baked into the product itself. Lexalytics is used similarly to monitor sentiment as it relates to stock trading. In December 2014, Lexalytics announced the latest iteration to its sentiment analysis engine, Salience 6. Earlier that year Lexalytics acquired Semantria in a bid to appeal to a wider variety of business models. Created by former Lexalytics Marketing Director Oleg Rogynskyy, Semantria is a SaaS text mining service offered as an API and Excel based plugin that measures sentiment. The goal of the acquisition, which cost Lexalytics less than US$10 million, was to expand the customer base both within the United States and abroad with multilingual support. The engine that powers Semantria, Salience, is grounded in its deep learning ability. An example of this is its concept matrix, which allows Salience an understanding of concepts and relationship between concepts based on a detailed reading of the entire repository of Wikipedia. This matrix allows Salience to use Wikipedia for automatic categorization. Along with features like the concept matrix, Salience supports 16 international languages. The engine has earned Lexalytics a spot on EContent's “Top 100 Companies in the Digital Content Industry” List for 2014–2015. In September 2018, Lexalytics launched document data extraction market using natural language processing (NLP).
Read more →
The Ni1000 is an artificial neural network chip developed by Nestor Corporation and Intel, developed in the 1990s. It is Intel's second-generation neural network chip, but the first all-digital chip. The chip is aimed at image analysis applications– containing more than 3 million transistors – and can analyze 40,000 patterns per second. Prototypes running Nestor's OCR software in 1994 were capable of recognizing around 100 handwritten characters per second. The development was funded with money from DARPA and Office of Naval Research.
Read more →
Sparse principal component analysis (SPCA or sparse PCA) is a technique used in statistical analysis and, in particular, in the analysis of multivariate data sets. It extends the classic method of principal component analysis (PCA) for the reduction of dimensionality of data by introducing sparsity structures to the input variables. A particular disadvantage of ordinary PCA is that the principal components are usually linear combinations of all input variables. SPCA overcomes this disadvantage by finding components that are linear combinations of just a few input variables (SPCs). This means that some of the coefficients of the linear combinations defining the SPCs, called loadings, are equal to zero. The number of nonzero loadings is called the cardinality of the SPC. == Mathematical formulation == Consider a data matrix, X {\displaystyle X} , where each of the p {\displaystyle p} columns represent an input variable, and each of the n {\displaystyle n} rows represents an independent sample from data population. One assumes each column of X {\displaystyle X} has mean zero, otherwise one can subtract column-wise mean from each element of X {\displaystyle X} . Let Σ = 1 n − 1 X ⊤ X {\displaystyle \Sigma ={\frac {1}{n-1}}X^{\top }X} be the empirical covariance matrix of X {\displaystyle X} , which has dimension p × p {\displaystyle p\times p} . Given an integer k {\displaystyle k} with 1 ≤ k ≤ p {\displaystyle 1\leq k\leq p} , the sparse PCA problem can be formulated as maximizing the variance along a direction represented by vector v ∈ R p {\displaystyle v\in \mathbb {R} ^{p}} while constraining its cardinality: max v T Σ v subject to ‖ v ‖ 2 = 1 ‖ v ‖ 0 ≤ k . {\displaystyle {\begin{aligned}\max \quad &v^{T}\Sigma v\\{\text{subject to}}\quad &\left\Vert v\right\Vert _{2}=1\\&\left\Vert v\right\Vert _{0}\leq k.\end{aligned}}} Eq. 1 The first constraint specifies that v is a unit vector. In the second constraint, ‖ v ‖ 0 {\displaystyle \left\Vert v\right\Vert _{0}} represents the ℓ 0 {\displaystyle \ell _{0}} pseudo-norm of v, which is defined as the number of its non-zero components. So the second constraint specifies that the number of non-zero components in v is less than or equal to k, which is typically an integer that is much smaller than dimension p. The optimal value of Eq. 1 is known as the k-sparse largest eigenvalue. If one takes k=p, the problem reduces to the ordinary PCA, and the optimal value becomes the largest eigenvalue of covariance matrix Σ. After finding the optimal solution v, one deflates Σ to obtain a new matrix Σ 1 = Σ − ( v T Σ v ) v v T , {\displaystyle \Sigma _{1}=\Sigma -(v^{T}\Sigma v)vv^{T},} and iterate this process to obtain further principal components. However, unlike PCA, sparse PCA cannot guarantee that different principal components are orthogonal. In order to achieve orthogonality, additional constraints must be enforced. The following equivalent definition is in matrix form. Let V {\displaystyle V} be a p×p symmetric matrix, one can rewrite the sparse PCA problem as max T r ( Σ V ) subject to T r ( V ) = 1 ‖ V ‖ 0 ≤ k 2 R a n k ( V ) = 1 , V ⪰ 0. {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\Vert V\Vert _{0}\leq k^{2}\\&Rank(V)=1,V\succeq 0.\end{aligned}}} Eq. 2 Tr is the matrix trace, and ‖ V ‖ 0 {\displaystyle \Vert V\Vert _{0}} represents the non-zero elements in matrix V. The last line specifies that V has matrix rank one and is positive semidefinite. The last line means that one has V = v v T {\displaystyle V=vv^{T}} , so Eq. 2 is equivalent to Eq. 1. Moreover, the rank constraint in this formulation is actually redundant, and therefore sparse PCA can be cast as the following mixed-integer semidefinite program max T r ( Σ V ) subject to T r ( V ) = 1 | V i , i | ≤ z i , ∀ i ∈ { 1 , . . . , p } , | V i , j | ≤ 1 2 z i , ∀ i , j ∈ { 1 , . . . , p } : i ≠ j , V ⪰ 0 , z ∈ { 0 , 1 } p , ∑ i z i ≤ k {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\vert V_{i,i}\vert \leq z_{i},\forall i\in \{1,...,p\},\vert V_{i,j}\vert \leq {\frac {1}{2}}z_{i},\forall i,j\in \{1,...,p\}:i\neq j,\\&V\succeq 0,z\in \{0,1\}^{p},\sum _{i}z_{i}\leq k\end{aligned}}} Eq. 3 Because of the cardinality constraint, the maximization problem is hard to solve exactly, especially when dimension p is high. In fact, the sparse PCA problem in Eq. 1 is NP-hard in the strong sense. == Computational considerations == As most sparse problems, variable selection in SPCA is a computationally intractable non-convex NP-hard problem, therefore greedy sub-optimal algorithms are often employed to find solutions. Note also that SPCA introduces hyperparameters quantifying in what capacity large parameter values are penalized. These might need tuning to achieve satisfactory performance, thereby adding to the total computational cost. == Algorithms for SPCA == Several alternative approaches (of Eq. 1) have been proposed, including a regression framework, a penalized matrix decomposition framework, a convex relaxation/semidefinite programming framework, a generalized power method framework an alternating maximization framework forward-backward greedy search and exact methods using branch-and-bound techniques, a certifiably optimal branch-and-bound approach Bayesian formulation framework. A certifiably optimal mixed-integer semidefinite branch-and-cut approach The methodological and theoretical developments of Sparse PCA as well as its applications in scientific studies are recently reviewed in a survey paper. === Notes on Semidefinite Programming Relaxation === It has been proposed that sparse PCA can be approximated by semidefinite programming (SDP). If one drops the rank constraint and relaxes the cardinality constraint by a 1-norm convex constraint, one gets a semidefinite programming relaxation, which can be solved efficiently in polynomial time: max T r ( Σ V ) subject to T r ( V ) = 1 1 T | V | 1 ≤ k V ⪰ 0. {\displaystyle {\begin{aligned}\max \quad &Tr(\Sigma V)\\{\text{subject to}}\quad &Tr(V)=1\\&\mathbf {1} ^{T}|V|\mathbf {1} \leq k\\&V\succeq 0.\end{aligned}}} Eq. 3 In the second constraint, 1 {\displaystyle \mathbf {1} } is a p×1 vector of ones, and |V| is the matrix whose elements are the absolute values of the elements of V. The optimal solution V {\displaystyle V} to the relaxed problem Eq. 3 is not guaranteed to have rank one. In that case, V {\displaystyle V} can be truncated to retain only the dominant eigenvector. While the semidefinite program does not scale beyond n=300 covariates, it has been shown that a second-order cone relaxation of the semidefinite relaxation is almost as tight and successfully solves problems with n=1000s of covariates == Applications == === Financial Data Analysis === Suppose ordinary PCA is applied to a dataset where each input variable represents a different asset, it may generate principal components that are weighted combination of all the assets. In contrast, sparse PCA would produce principal components that are weighted combination of only a few input assets, so one can easily interpret its meaning. Furthermore, if one uses a trading strategy based on these principal components, fewer assets imply less transaction costs. === Biology === Consider a dataset where each input variable corresponds to a specific gene. Sparse PCA can produce a principal component that involves only a few genes, so researchers can focus on these specific genes for further analysis. === High-dimensional Hypothesis Testing === Contemporary datasets often have the number of input variables ( p {\displaystyle p} ) comparable with or even much larger than the number of samples ( n {\displaystyle n} ). It has been shown that if p / n {\displaystyle p/n} does not converge to zero, the classical PCA is not consistent. In other words, if we let k = p {\displaystyle k=p} in Eq. 1, then the optimal value does not converge to the largest eigenvalue of data population when the sample size n → ∞ {\displaystyle n\rightarrow \infty } , and the optimal solution does not converge to the direction of maximum variance. But sparse PCA can retain consistency even if p ≫ n . {\displaystyle p\gg n.} The k-sparse largest eigenvalue (the optimal value of Eq. 1) can be used to discriminate an isometric model, where every direction has the same variance, from a spiked covariance model in high-dimensional setting. Consider a hypothesis test where the null hypothesis specifies that data X {\displaystyle X} are generated from a multivariate normal distribution with mean 0 and covariance equal to an identity matrix, and the alternative hypothesis specifies that data X {\displaystyle X} is generated from a spiked model with signal strength θ {\displaystyle \theta } : H 0 : X ∼ N ( 0 , I p ) , H 1 : X ∼ N ( 0 , I p + θ v v T ) , {\displaystyle H_{0}:X\sim N(0,I_{p}),\quad H_{1}:X\sim N(0,I_{p}+\theta vv^{T}),} where v ∈ R p {\displaystyle v\in \mathbb {R} ^{p}
Read more →