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Wetware computer

A wetware computer is an organic computer (which can also be known as an artificial organic brain or a neurocomputer) composed of organic material "wetware" such as "living" neurons. Wetware computers composed of neurons are different than conventional computers because they use biological materials, and offer the possibility of substantially more energy-efficient computing. While a wetware computer is still largely conceptual, there has been limited success with construction and prototyping, which has acted as a proof of the concept's realistic application to computing in the future. The most notable prototypes have stemmed from the research completed by biological engineer William Ditto during his time at the Georgia Institute of Technology. His work constructing a simple neurocomputer capable of basic addition from leech neurons in 1999 was a significant discovery for the concept. This research was a primary example driving interest in creating these artificially constructed, but still organic brains. == Origins and theoretical foundations == The term wetware came from cyberpunk fiction, notably through Gibson's Neuromancer, but was quickly taken up in scientific literature to explain computation by biological material. Theories of early biological computation borrowed from Alan Turing's morphogenesis model, which showed that chemical interactions could produce complex patterns without centralized control. Hopfield's associative memory networks also provided a foundation for biological information systems with fault tolerance and self-organization. == Major characteristics and processes == Biological wetware systems demonstrate dynamic reconfigurability underpinned by neuroplasticity and enable continuous learning and adaptation. Reaction-diffusion-based computing and molecular logic gates allow spatially parallel information processing unachievable in conventional systems. These systems also show fault tolerance and self-repair at the cellular and network level. The development of cerebral organoids—miniature lab-grown brains—demonstrates spontaneous learning behavior and suggests biological tissue as a viable computational substrate. == Overview == The concept of wetware is an application of specific interest to the field of computer manufacturing. Moore's law, which states that the number of transistors which can be placed on a silicon chip is doubled roughly every two years, has acted as a goal for the industry for decades, but as the size of computers continues to decrease, the ability to meet this goal has become more difficult, threatening to reach a plateau. Due to the difficulty in reducing the size of computers because of size limitations of transistors and integrated circuits, wetware provides an unconventional alternative. A wetware computer composed of neurons is an ideal concept because, unlike conventional materials which operate in binary (on/off), a neuron can shift between thousands of states, constantly altering its chemical conformation, and redirecting electrical pulses through over 200,000 channels in any of its many synaptic connections. Because of this large difference in the possible settings for any one neuron, compared to the binary limitations of conventional computers, the space limitations are far fewer. == Background == The concept of wetware is distinct and unconventional and draws slight resonance with both hardware and software from conventional computers. While hardware is understood as the physical architecture of traditional computational devices, comprising integrated circuits and supporting infrastructure, software represents the encoded architecture of storage and instructions. Wetware is a separate concept that uses the formation of organic molecules, mostly complex cellular structures (such as neurons), to create a computational device such as a computer. In wetware, the ideas of hardware and software are intertwined and interdependent. The molecular and chemical composition of the organic or biological structure would represent not only the physical structure of the wetware but also the software, being continually reprogrammed by the discrete shifts in electrical pulses and chemical concentration gradients as the molecules change their structures to communicate signals. The responsiveness of a cell, proteins, and molecules to changing conformations, both within their structures and around them, ties the idea of internal programming and external structure together in a way that is alien to the current model of conventional computer architecture. The structure of wetware represents a model where the external structure and internal programming are interdependent and unified; meaning that changes to the programming or internal communication between molecules of the device would represent a physical change in the structure. The dynamic nature of wetware borrows from the function of complex cellular structures in biological organisms. The combination of "hardware" and "software" into one dynamic, and interdependent system which uses organic molecules and complexes to create an unconventional model for computational devices is a specific example of applied biorobotics. === The cell as a model of wetware === Cells in many ways can be seen as their form of naturally occurring wetware, similar to the concept that the human brain is the preexisting model system for complex wetware. In his book Wetware: A Computer in Every Living Cell (2009) Dennis Bray explains his theory that cells, which are the most basic form of life, are just a highly complex computational structure, like a computer. To simplify one of his arguments a cell can be seen as a type of computer, using its structured architecture. In this architecture, much like a traditional computer, many smaller components operate in tandem to receive input, process the information, and compute an output. In an overly simplified, non-technical analysis, cellular function can be broken into the following components: Information and instructions for execution are stored as DNA in the cell, RNA acts as a source for distinctly encoded input, processed by ribosomes and other transcription factors to access and process the DNA and to output a protein. Bray's argument in favor of viewing cells and cellular structures as models of natural computational devices is important when considering the more applied theories of wetware to biorobotics. === Biorobotics === Wetware and biorobotics are closely related concepts, which both borrow from similar overall principles. A biorobotic structure can be defined as a system modeled from a preexisting organic complex or model such as cells (neurons) or more complex structures like organs (brain) or whole organisms. Unlike wetware, the concept of biorobotics is not always a system composed of organic molecules, but instead could be composed of conventional material which is designed and assembled in a structure similar or derived from a biological model. Biorobotics have many applications and are used to address the challenges of conventional computer architecture. Conceptually, designing a program, robot, or computational device after a preexisting biological model such as a cell, or even a whole organism, provides the engineer or programmer the benefits of incorporating into the structure the evolutionary advantages of the model. == Effects on users == Wetware technologies such as BCIs and neuromorphic chips offer new possibilities for user autonomy. For those with disabilities, such systems could restore motor or sensory functions and enhance quality of life. However, these technologies raise ethical questions: cognitive privacy, consent over biological data, and risk of exploitation. Without proper oversight, wetware technologies may also widen inequality, favoring those with access to cognitive enhancements. Open governance frameworks and ethical AI design grounded in neuro ethics will be essential. With the development of wetware devices, disparities in access could exacerbate social inequalities, benefiting those who have resources to enhance cognitive or physical abilities. It is necessary to create strong ethical frameworks, inclusive development practices, and open systems of governance to reduce risks and make sure that wetware advances are beneficial to all segments of society. == Applications and goals == === Basic neurocomputer composed of leech neurons === In 1999 William Ditto and his team of researchers at Georgia Institute of Technology and Emory University created a basic form of a wetware computer capable of simple addition by harnessing leech neurons. Leeches were used as a model organism due to the large size of their neuron, and the ease associated with their collection and manipulation. However, these results have never been published in a peer-reviewed journal, prompting questions about the validity of the claims. The computer was able to complete basic addition through electrical probes
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Policy gradient method

Policy gradient methods are a class of reinforcement learning algorithms and a sub-class of policy optimization methods. Unlike value-based methods which learn a value function to derive a policy, policy optimization methods directly learn a policy function π {\displaystyle \pi } that selects actions without consulting a value function. For policy gradient to apply, the policy function π θ {\displaystyle \pi _{\theta }} is parameterized by a differentiable parameter θ {\displaystyle \theta } . == Overview == In policy-based RL, the actor is a parameterized policy function π θ {\displaystyle \pi _{\theta }} , where θ {\displaystyle \theta } are the parameters of the actor. The actor takes as argument the state of the environment s {\displaystyle s} and produces a probability distribution π θ ( ⋅ ∣ s ) {\displaystyle \pi _{\theta }(\cdot \mid s)} . If the action space is discrete, then ∑ a π θ ( a ∣ s ) = 1 {\displaystyle \sum _{a}\pi _{\theta }(a\mid s)=1} . If the action space is continuous, then ∫ a π θ ( a ∣ s ) d a = 1 {\displaystyle \int _{a}\pi _{\theta }(a\mid s)\mathrm {d} a=1} . The goal of policy optimization is to find some θ {\displaystyle \theta } that maximizes the expected episodic reward J ( θ ) {\displaystyle J(\theta )} : J ( θ ) = E π θ [ ∑ t = 0 T γ t R t | S 0 = s 0 ] {\displaystyle J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\gamma ^{t}R_{t}{\Big |}S_{0}=s_{0}\right]} where γ {\displaystyle \gamma } is the discount factor, R t {\displaystyle R_{t}} is the reward at step t {\displaystyle t} , s 0 {\displaystyle s_{0}} is the starting state, and T {\displaystyle T} is the time-horizon (which can be infinite). The policy gradient is defined as ∇ θ J ( θ ) {\displaystyle \nabla _{\theta }J(\theta )} . Different policy gradient methods stochastically estimate the policy gradient in different ways. The goal of any policy gradient method is to iteratively maximize J ( θ ) {\displaystyle J(\theta )} by gradient ascent. Since the key part of any policy gradient method is the stochastic estimation of the policy gradient, they are also studied under the title of "Monte Carlo gradient estimation". == REINFORCE == === Policy gradient === The REINFORCE algorithm, introduced by Ronald J. Williams in 1992, was the first policy gradient method. It is based on the identity for the policy gradient ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ t = 0 T ( γ t R t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\;\sum _{t=0}^{T}(\gamma ^{t}R_{t}){\Big |}S_{0}=s_{0}\right]} which can be improved via the "causality trick" ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t ∣ S t ) ∑ τ = t T ( γ τ R τ ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau }){\Big |}S_{0}=s_{0}\right]} Thus, we have an unbiased estimator of the policy gradient: ∇ θ J ( θ ) ≈ 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ − t R τ , n ) ] {\displaystyle \nabla _{\theta }J(\theta )\approx {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau -t}R_{\tau ,n})\right]} where the index n {\displaystyle n} ranges over N {\displaystyle N} rollout trajectories using the policy π θ {\displaystyle \pi _{\theta }} . The score function ∇ θ ln ⁡ π θ ( A t ∣ S t ) {\displaystyle \nabla _{\theta }\ln \pi _{\theta }(A_{t}\mid S_{t})} can be interpreted as the direction in the parameter space that increases the probability of taking action A t {\displaystyle A_{t}} in state S t {\displaystyle S_{t}} . The policy gradient, then, is a weighted average of all possible directions to increase the probability of taking any action in any state, but weighted by reward signals, so that if taking a certain action in a certain state is associated with high reward, then that direction would be highly reinforced, and vice versa. === Algorithm === The REINFORCE algorithm is a loop: Rollout N {\displaystyle N} trajectories in the environment, using π θ t {\displaystyle \pi _{\theta _{t}}} as the policy function. Compute the policy gradient estimation: g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n ∣ S t , n ) ∑ τ = t T ( γ τ R τ , n ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}\mid S_{t,n})\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})\right]} Update the policy by gradient ascent: θ i + 1 ← θ i + α i g i {\displaystyle \theta _{i+1}\leftarrow \theta _{i}+\alpha _{i}g_{i}} Here, α i {\displaystyle \alpha _{i}} is the learning rate at update step i {\displaystyle i} . == Variance reduction == REINFORCE is an on-policy algorithm, meaning that the trajectories used for the update must be sampled from the current policy π θ {\displaystyle \pi _{\theta }} . This can lead to high variance in the updates, as the returns R ( τ ) {\displaystyle R(\tau )} can vary significantly between trajectories. Many variants of REINFORCE have been introduced, under the title of variance reduction. === REINFORCE with baseline === A common way for reducing variance is the REINFORCE with baseline algorithm, based on the following identity: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − b ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-b(S_{t})\right){\Big |}S_{0}=s_{0}\right]} for any function b : States → R {\displaystyle b:{\text{States}}\to \mathbb {R} } . This can be proven by applying the previous lemma. The algorithm uses the modified gradient estimator g i ← 1 N ∑ n = 1 N [ ∑ t = 0 T ∇ θ t ln ⁡ π θ ( A t , n | S t , n ) ( ∑ τ = t T ( γ τ R τ , n ) − b i ( S t , n ) ) ] {\displaystyle g_{i}\leftarrow {\frac {1}{N}}\sum _{n=1}^{N}\left[\sum _{t=0}^{T}\nabla _{\theta _{t}}\ln \pi _{\theta }(A_{t,n}|S_{t,n})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau ,n})-b_{i}(S_{t,n})\right)\right]} and the original REINFORCE algorithm is the special case where b i ≡ 0 {\displaystyle b_{i}\equiv 0} . === Actor-critic methods === If b i {\textstyle b_{i}} is chosen well, such that b i ( S t ) ≈ ∑ τ = t T ( γ τ R τ ) = γ t V π θ i ( S t ) {\textstyle b_{i}(S_{t})\approx \sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })=\gamma ^{t}V^{\pi _{\theta _{i}}}(S_{t})} , this could significantly decrease variance in the gradient estimation. That is, the baseline should be as close to the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} as possible, approaching the ideal of: ∇ θ J ( θ ) = E π θ [ ∑ t = 0 T ∇ θ ln ⁡ π θ ( A t | S t ) ( ∑ τ = t T ( γ τ R τ ) − γ t V π θ ( S t ) ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=\mathbb {E} _{\pi _{\theta }}\left[\sum _{t=0}^{T}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\left(\sum _{\tau =t}^{T}(\gamma ^{\tau }R_{\tau })-\gamma ^{t}V^{\pi _{\theta }}(S_{t})\right){\Big |}S_{0}=s_{0}\right]} Note that, as the policy π θ t {\displaystyle \pi _{\theta _{t}}} updates, the value function V π θ i ( S t ) {\displaystyle V^{\pi _{\theta _{i}}}(S_{t})} updates as well, so the baseline should also be updated. One common approach is to train a separate function that estimates the value function, and use that as the baseline. This is one of the actor-critic methods, where the policy function is the actor and the value function is the critic. The Q-function Q π {\displaystyle Q^{\pi }} can also be used as the critic, since ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Q π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot Q^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} by a similar argument using the tower law. Subtracting the value function as a baseline, we find that the advantage function A π ( S , A ) = Q π ( S , A ) − V π ( S ) {\displaystyle A^{\pi }(S,A)=Q^{\pi }(S,A)-V^{\pi }(S)} can be used as the critic as well: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T γ t ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ A π θ ( S t , A t ) | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\sum _{0\leq t\leq T}\gamma ^{t}\nabla _{\theta }\ln \pi _{\theta }(A_{t}|S_{t})\cdot A^{\pi _{\theta }}(S_{t},A_{t}){\Big |}S_{0}=s_{0}\right]} In summary, there are many unbiased estimators for ∇ θ J θ {\textstyle \nabla _{\theta }J_{\theta }} , all in the form of: ∇ θ J ( θ ) = E π θ [ ∑ 0 ≤ t ≤ T ∇ θ ln ⁡ π θ ( A t | S t ) ⋅ Ψ t | S 0 = s 0 ] {\displaystyle \nabla _{\theta }J(\theta )=E_{\pi _{\theta }}\left[\su
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Teaching dimension

In computational learning theory, the teaching dimension of a concept class C is defined to be max c ∈ C { w C ( c ) } {\displaystyle \max _{c\in C}\{w_{C}(c)\}} , where w C ( c ) {\displaystyle {w_{C}(c)}} is the minimum size of a witness set for c in C. Intuitively, this measures the number of instances that are needed to identify a concept in the class, using supervised learning with examples provided by a helpful teacher who is trying to convey the concept as succinctly as possible. This definition was formulated in 1995 by Sally Goldman and Michael Kearns, based on earlier work by Goldman, Ron Rivest, and Robert Schapire. The teaching dimension of a finite concept class can be used to give a lower and an upper bound on the membership query cost of the concept class. In Stasys Jukna's book "Extremal Combinatorics", a lower bound is given for the teaching dimension in general: Let C be a concept class over a finite domain X. If the size of C is greater than 2 k ( | X | k ) , {\displaystyle 2^{k}{|X| \choose k},} then the teaching dimension of C is greater than k. However, there are more specific teaching models that make assumptions about teacher or learner, and can get lower values for the teaching dimension. For instance, several models are the classical teaching (CT) model, the optimal teacher (OT) model, recursive teaching (RT), preference-based teaching (PBT), and non-clashing teaching (NCT).
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Sum of absolute differences

In digital image processing, the sum of absolute differences (SAD) is a measure of the similarity between image blocks. It is calculated by taking the absolute difference between each pixel in the original block and the corresponding pixel in the block being used for comparison. These differences are summed to create a simple metric of block similarity, the L1 norm of the difference image or Manhattan distance between two image blocks. The sum of absolute differences may be used for a variety of purposes, such as object recognition, the generation of disparity maps for stereo images, and motion estimation for video compression. == Example == This example uses the sum of absolute differences to identify which part of a search image is most similar to a template image. In this example, the template image is 3 by 3 pixels in size, while the search image is 3 by 5 pixels in size. Each pixel is represented by a single integer from 0 to 9. Template Search image 2 5 5 2 7 5 8 6 4 0 7 1 7 4 2 7 7 5 9 8 4 6 8 5 There are exactly three unique locations within the search image where the template may fit: the left side of the image, the center of the image, and the right side of the image. To calculate the SAD values, the absolute value of the difference between each corresponding pair of pixels is used: the difference between 2 and 2 is 0, 4 and 1 is 3, 7 and 8 is 1, and so forth. Calculating the values of the absolute differences for each pixel, for the three possible template locations, gives the following: Left Center Right 0 2 0 5 0 3 3 3 1 3 7 3 3 4 5 0 2 0 1 1 3 3 1 1 1 3 4 For each of these three image patches, the 9 absolute differences are added together, giving SAD values of 20, 25, and 17, respectively. From these SAD values, it could be asserted that the right side of the search image is the most similar to the template image, because it has the lowest sum of absolute differences as compared to the other two locations. == Comparison to other metrics == === Object recognition === The sum of absolute differences provides a simple way to automate the searching for objects inside an image, but may be unreliable due to the effects of contextual factors such as changes in lighting, color, viewing direction, size, or shape. The SAD may be used in conjunction with other object recognition methods, such as edge detection, to improve the reliability of results. === Video compression === SAD is an extremely fast metric due to its simplicity; it is effectively the simplest possible metric that takes into account every pixel in a block. Therefore, it is very effective for a wide motion search of many different blocks. SAD is also easily parallelizable since it analyzes each pixel separately, making it easily implementable with such instructions as ARM NEON or x86 SSE2. For example, SSE has packed sum of absolute differences instruction (PSADBW) specifically for this purpose. Once candidate blocks are found, the final refinement of the motion estimation process is often done with other slower but more accurate metrics, which better take into account human perception. These include the sum of absolute transformed differences (SATD), the sum of squared differences (SSD), and rate–distortion optimization.
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Text-to-image personalization

Text-to-Image personalization is a task in deep learning for computer graphics that augments pre-trained text-to-image generative models. In this task, a generative model that was trained on large-scale data (usually a foundation model), is adapted such that it can generate images of novel, user-provided concepts. These concepts are typically unseen during training, and may represent specific objects (such as the user's pet) or more abstract categories (new artistic style or object relations). Text-to-Image personalization methods typically bind the novel (personal) concept to new words in the vocabulary of the model. These words can then be used in future prompts to invoke the concept for subject-driven generation, inpainting, style transfer and even to correct biases in the model. To do so, models either optimize word-embeddings, fine-tune the generative model itself, or employ a mixture of both approaches. == Technology == Text-to-Image personalization was first proposed during August 2022 by two concurrent works, Textual Inversion and DreamBooth. In both cases, a user provides a few images (typically 3–5) of a concept, like their own dog, together with a coarse descriptor of the concept class (like the word "dog"). The model then learns to represent the subject through a reconstruction based objective, where prompts referring to the subject are expected to reconstruct images from the training set. In Textual Inversion, the personalized concepts are introduced into the text-to-image model by adding new words to the vocabulary of the model. Typical text-to-image models represent words (and sometimes parts-of-words) as tokens, or indices in a predefined dictionary. During generation, an input prompt is converted into such tokens, each of which is converted into a ‘word-embedding’: a continuous vector representation which is learned for each token as part of the model's training. Textual Inversion proposes to optimize a new word-embedding vector for representing the novel concept. This new embedding vector can then be assigned to a user-chosen string, and invoked whenever the user's prompt contains this string. In DreamBooth, rather than optimizing a new word vector, the full generative model itself is fine-tuned. The user first selects an existing token, typically one which rarely appears in prompts. The subject itself is then represented by a string containing this token, followed by a coarse descriptor of the subject's class. A prompt describing the subject will then take the form: "A photo of " (e.g. "a photo of sks cat" when learning to represent a specific cat). The text-to-image model is then tuned so that prompts of this form will generate images of the subject. == Textual Inversion == The key idea in Textual Inversion is to add a new term to the vocabulary of the diffusion model that corresponds to the new (personalized) concept. Textual Inversion operates by inverting the concepts into new pseudo-words within the textual embedding space of a pre-trained text-to-image model. These pseudo-words can be injected into new scenes using simple natural language descriptions, allowing for simple and intuitive modifications. The method allows a user to leverage multi-modal information — using a text-driven interface for ease of editing, but providing visual cues when approaching the limits of natural language. The resulting model is extremely light-weight per concept: only 1K long, but succeeds to encode detailed visual properties of the concept. == Extensions == Several approaches were proposed to refine and improve over the original methods. These include the following. Low-rank Adaptation (LoRA) - an adapter-based technique for efficient finetuning of models. In the case of text-to-image models, LoRA is typically used to modify the cross-attention layers of a diffusion model. Perfusion - a low rank update method that also locks the activations of the key matrix in the diffusion model's cross attention layers to the concept's coarse class. Extended Textual Inversion - a technique that learns an individual word embedding for each layer in the diffusion model's denoising network. Encoder-based methods that use another neural network to quickly personalize a model == Challenges and limitations == Text-to-image personalization methods must contend with several challenges. At their core is the goal of achieving high-fidelity to the personal concept while maintaining high alignment between novel prompts containing the subject, and the generated images (typically referred to as ‘editability’). Another challenge that personalization methods must contend with is memory requirements. Initial implementations of personalization methods required more than 20 Gigabytes of GPU memory, and more recent approaches have reported requirements of more than 40 Gigabytes. However, optimizations such as Flash Attention have since reduced this requirement considerably. Approaches that tune the entire generative model may also create checkpoints that are several gigabytes in size, making it difficult to share or store many models. Embedding based approaches require only a few kilobytes, but typically struggle to preserve identity while maintaining editability. More recent approaches have proposed hybrid tuning goals which optimize both an embedding and a subset of network weights. These can reduce storage requirements to as little as 100 Kilobytes while achieving quality comparable to full tuning methods. Finally, optimization processes can be lengthy, requiring several minutes of tuning for each novel concept. Encoder and quick-tuning methods aim to reduce this to seconds or less.
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Generalized blockmodeling

In generalized blockmodeling, the blockmodeling is done by "the translation of an equivalence type into a set of permitted block types", which differs from the conventional blockmodeling, which is using the indirect approach. It's a special instance of the direct blockmodeling approach. Generalized blockmodeling was introduced in 1994 by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj. == Definition == Generalized blockmodeling approach is a direct one, "where the optimal partition(s) is (are) identified based on minimal values of a compatible criterion function defined by the difference between empirical blocks and corresponding ideal blocks". At the same time, the much broader set of block types is introduced (while in conventional blockmodeling only certain types are used). The conventional blockmodeling is inductive due to nonspecification of neither the clusters or the location of block types, while in generalized blockmodeling the blockmodel is specified with more detail than just the permition of certain block types (e.g., prespecification). Further, it's possible to define departures from the permitted (ideal) blocktype, using criterion function. Using local optimization procedure, firstly the initial clustering (with specified number of clusters is done, based on random creation. How the clusters are neighboring to each other, is based on two transformations: 1) a vertex is moved from one to another cluster or 2) a pair of vertices is interchanged between two different clusters. This process of transformation steps is repeated many times, until only the best fitting partitions (with the minimized value of the criterion function) are kept as blockmodels for the future exploration of the network. Different types of generalized blockmodeling are: generalized binary blockmodeling, generalized valued blockmodeling and generalized homogeneity blockmodeling. == Benefits == According to Patrick Doreian, the benefits of generalized blockmodeling, are as follows: usage of explicit criterion function, compatible with a given type of equivalence, results to in-built measure of fit, which is integral to the establishment of the blockmodels (in conventional blockmodeling, there is no compelling and coherent measures of fit); partitions, based on generalized blockmodeling, regularly outperform and never perform less well than the partitions, based on conventional approach; with generalized blockmodeling it's possible to specify new types of blockmodels; this potentially unlimited set of new block types also results in permittion of inclusion of substantively driven blockmodels; in generalized blockmodeling, the specification of the block types and the location of some of them in the blockmodel is possible; researcher can speficy which (pair of) vertices must be (not) clustered together; this approach also allows the imposition of penalties, resulting into identification of empirical null blocks without inconsistencies with a corresponding ideal null block. == Problems == According to Doreian, the problems of generalized blockmodeling, are as follows: unknown sensitivity to particular data features, examination of boundary problems, computationally burdensome, which results in a constraint regarding practical network size (generalized blockmodeling is thus primarily used to analyse smaller networks (below 100 units)), identifying structure from incomplete network information, most of generalized blockmodeling is based on binary networks, but there is also development in the field of valued networks, criterion function is minimized for a specified blockmodel, with results in issues of evaluating statistically, based on the structural data alone, problems regarding three dimensional network data, problems regarding the evolution of fundamental network structure. == Book == The book with the same title, Generalized blockmodeling, written by Patrick Doreian, Vladimir Batagelj and Anuška Ferligoj, was in 2007 awarded the Harrison White Outstanding Book Award by the Mathematical Sociology Section of American Sociological Association.
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Clustering illusion

The clustering illusion is the tendency to erroneously consider the inevitable "streaks" or "clusters" arising in small samples from random distributions to be non-random. The illusion is caused by a human tendency to underpredict the amount of variability likely to appear in a small sample of random or pseudorandom data. Thomas Gilovich, an early author on the subject, argued that the effect occurs for different types of random dispersions. Some might perceive patterns in stock market price fluctuations over time, or clusters in two-dimensional data such as the locations of impact of World War II V-1 flying bombs on maps of London. Although Londoners developed specific theories about the pattern of impacts within London, a statistical analysis by R. D. Clarke originally published in 1946 showed that the impacts of V-2 rockets on London were a close fit to a random distribution. == Similar biases == Using this cognitive bias in causal reasoning may result in the Texas sharpshooter fallacy, in which differences in data are ignored and similarities are overemphasized. More general forms of erroneous pattern recognition are pareidolia and apophenia. Related biases are the illusion of control which the clustering illusion could contribute to, and insensitivity to sample size in which people don't expect greater variation in smaller samples. A different cognitive bias involving misunderstanding of chance streams is the gambler's fallacy. == Possible causes == Daniel Kahneman and Amos Tversky explained this kind of misprediction as being caused by the representativeness heuristic (which itself they also first proposed).
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Semantic mapping (statistics)

Semantic mapping (SM) is a statistical method for dimensionality reduction (the transformation of data from a high-dimensional space into a low-dimensional space). SM can be used in a set of multidimensional vectors of features to extract a few new features that preserves the main data characteristics. SM performs dimensionality reduction by clustering the original features in semantic clusters and combining features mapped in the same cluster to generate an extracted feature. Given a data set, this method constructs a projection matrix that can be used to map a data element from a high-dimensional space into a reduced dimensional space. SM can be applied in construction of text mining and information retrieval systems, as well as systems managing vectors of high dimensionality. SM is an alternative to random mapping, principal components analysis and latent semantic indexing methods.
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Model compression

Model compression is a machine learning technique for reducing the size of trained models. Large models can achieve high accuracy, but often at the cost of significant resource requirements. Compression techniques aim to compress models without significant performance reduction. Smaller models require less storage space, and consume less memory and compute during inference. Compressed models enable deployment on resource-constrained devices such as smartphones, embedded systems, edge computing devices, and consumer electronics computers. Efficient inference is also valuable for large corporations that serve large model inference over an API, allowing them to reduce computational costs and improve response times for users. Model compression is not to be confused with knowledge distillation, in which a smaller "student" model is trained to imitate the input-output behavior of a larger "teacher" model (as opposed to using the "teacher"'s trained parameters or the "teacher"'s training targets). == Techniques == Several techniques are employed for model compression. === Pruning === Pruning sparsifies a large model by setting some parameters to exactly zero. This effectively reduces the number of parameters. This allows the use of sparse matrix operations, which are faster than dense matrix operations. Pruning criteria can be based on magnitudes of parameters, the statistical pattern of neural activations, Hessian values, etc. === Quantization === Quantization reduces the numerical precision of weights and activations. For example, instead of storing weights as 32-bit floating-point numbers, they can be represented using 8-bit integers. Low-precision parameters take up less space, and takes less compute to perform arithmetic with. It is also possible to quantize some parameters more aggressively than others, so for example, a less important parameter can have 8-bit precision while another, more important parameter, can have 16-bit precision. Inference with such models requires mixed-precision arithmetic. Quantized models can also be used during training (rather than after training). PyTorch implements automatic mixed-precision (AMP), which performs autocasting, gradient scaling, and loss scaling. === Low-rank factorization === Weight matrices can be approximated by low-rank matrices. Let W {\displaystyle W} be a weight matrix of shape m × n {\displaystyle m\times n} . A low-rank approximation is W ≈ U V T {\displaystyle W\approx UV^{T}} , where U {\displaystyle U} and V {\displaystyle V} are matrices of shapes m × k , n × k {\displaystyle m\times k,n\times k} . When k {\displaystyle k} is small, this both reduces the number of parameters needed to represent W {\displaystyle W} approximately, and accelerates matrix multiplication by W {\displaystyle W} . Low-rank approximations can be found by singular value decomposition (SVD). The choice of rank for each weight matrix is a hyperparameter, and jointly optimized as a mixed discrete-continuous optimization problem. The rank of weight matrices may also be pruned after training, taking into account the effect of activation functions like ReLU on the implicit rank of the weight matrices. == Training == Model compression may be decoupled from training, that is, a model is first trained without regard for how it might be compressed, then it is compressed. However, it may also be combined with training. The "train big, then compress" method trains a large model for a small number of training steps (less than it would be if it were trained to convergence), then heavily compress the model. It is found that at the same compute budget, this method results in a better model than lightly compressed, small models. In Deep Compression, the compression has three steps. First loop (pruning): prune all weights lower than a threshold, then finetune the network, then prune again, etc. Second loop (quantization): cluster weights, then enforce weight sharing among all weights in each cluster, then finetune the network, then cluster again, etc. Third step: Use Huffman coding to losslessly compress the model. The SqueezeNet paper reported that Deep Compression achieved a compression ratio of 35 on AlexNet, and a ratio of ~10 on SqueezeNets.
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Gaussian process emulator

In statistics, Gaussian process emulator is one name for a general type of statistical model that has been used in contexts where the problem is to make maximum use of the outputs of a complicated (often non-random) computer-based simulation model. Each run of the simulation model is computationally expensive and each run is based on many different controlling inputs. The variation of the outputs of the simulation model is expected to vary reasonably smoothly with the inputs, but in an unknown way. The overall analysis involves two models: the simulation model, or "simulator", and the statistical model, or "emulator", which notionally emulates the unknown outputs from the simulator. The Gaussian process emulator model treats the problem from the viewpoint of Bayesian statistics. In this approach, even though the output of the simulation model is fixed for any given set of inputs, the actual outputs are unknown unless the computer model is run and hence can be made the subject of a Bayesian analysis. The main element of the Gaussian process emulator model is that it models the outputs as a Gaussian process on a space that is defined by the model inputs. The model includes a description of the correlation or covariance of the outputs, which enables the model to encompass the idea that differences in the output will be small if there are only small differences in the inputs.
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Bayesian hierarchical modeling

Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the posterior distribution of model parameters using the Bayesian method. The sub-models combine to form the hierarchical model, and Bayes' theorem is used to integrate them with the observed data and account for all the uncertainty that is present. This integration enables calculation of updated posterior over the (hyper)parameters, effectively updating prior beliefs in light of the observed data. Frequentist statistics may yield conclusions seemingly incompatible with those offered by Bayesian statistics due to the Bayesian treatment of the parameters as random variables and its use of subjective information in establishing assumptions on these parameters. As the approaches answer different questions the formal results are not technically contradictory but the two approaches disagree over which answer is relevant to particular applications. Bayesians argue that relevant information regarding decision-making and updating beliefs cannot be ignored and that hierarchical modeling has the potential to overrule classical methods in applications where respondents give multiple observational data. Moreover, the model has proven to be robust, with the posterior distribution less sensitive to the more flexible hierarchical priors. Hierarchical modeling, as its name implies, retains nested data structure, and is used when information is available at several different levels of observational units. For example, in epidemiological modeling to describe infection trajectories for multiple countries, observational units are countries, and each country has its own time-based profile of daily infected cases. In decline curve analysis to describe oil or gas production decline curve for multiple wells, observational units are oil or gas wells in a reservoir region, and each well has each own time-based profile of oil or gas production rates (usually, barrels per month). Hierarchical modeling is used to devise computation based strategies for multiparameter problems. == Philosophy == Statistical methods and models commonly involve multiple parameters that can be regarded as related or connected in such a way that the problem implies a dependence of the joint probability model for these parameters. Individual degrees of belief, expressed in the form of probabilities, come with uncertainty. Amidst this is the change of the degrees of belief over time. As was stated by Professor José M. Bernardo and Professor Adrian F. Smith, "The actuality of the learning process consists in the evolution of individual and subjective beliefs about the reality." These subjective probabilities are more directly involved in the mind rather than the physical probabilities. Hence, it is with this need of updating beliefs that Bayesians have formulated an alternative statistical model which takes into account the prior occurrence of a particular event. == Bayes' theorem == The assumed occurrence of a real-world event will typically modify preferences between certain options. This is done by modifying the degrees of belief attached, by an individual, to the events defining the options. Suppose in a study of the effectiveness of cardiac treatments, with the patients in hospital j having survival probability θ j {\displaystyle \theta _{j}} , the survival probability will be updated with the occurrence of y, the event in which a controversial serum is created which, as believed by some, increases survival in cardiac patients. In order to make updated probability statements about θ j {\displaystyle \theta _{j}} , given the occurrence of event y, we must begin with a model providing a joint probability distribution for θ j {\displaystyle \theta _{j}} and y. This can be written as a product of the two distributions that are often referred to as the prior distribution P ( θ ) {\displaystyle P(\theta )} and the sampling distribution P ( y ∣ θ ) {\displaystyle P(y\mid \theta )} respectively: P ( θ , y ) = P ( θ ) P ( y ∣ θ ) {\displaystyle P(\theta ,y)=P(\theta )P(y\mid \theta )} Using the basic property of conditional probability, the posterior distribution will yield: P ( θ ∣ y ) = P ( θ , y ) P ( y ) = P ( y ∣ θ ) P ( θ ) P ( y ) {\displaystyle P(\theta \mid y)={\frac {P(\theta ,y)}{P(y)}}={\frac {P(y\mid \theta )P(\theta )}{P(y)}}} This equation, showing the relationship between the conditional probability and the individual events, is known as Bayes' theorem. This simple expression encapsulates the technical core of Bayesian inference which aims to deconstruct the probability, P ( θ ∣ y ) {\displaystyle P(\theta \mid y)} , relative to solvable subsets of its supportive evidence. == Exchangeability == The usual starting point of a statistical analysis is the assumption that the n values y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} are exchangeable. If no information – other than data y – is available to distinguish any of the θ j {\displaystyle \theta _{j}} 's from any others, and no ordering or grouping of the parameters can be made, one must assume symmetry of prior distribution parameters. This symmetry is represented probabilistically by exchangeability. Generally, it is useful and appropriate to model data from an exchangeable distribution as independently and identically distributed, given some unknown parameter vector θ {\displaystyle \theta } , with distribution P ( θ ) {\displaystyle P(\theta )} . === Finite exchangeability === For a fixed number n, the set y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable if the joint probability P ( y 1 , y 2 , … , y n ) {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})} is invariant under permutations of the indices. That is, for every permutation π {\displaystyle \pi } or ( π 1 , π 2 , … , π n ) {\displaystyle (\pi _{1},\pi _{2},\ldots ,\pi _{n})} of (1, 2, …, n), P ( y 1 , y 2 , … , y n ) = P ( y π 1 , y π 2 , … , y π n ) . {\displaystyle P(y_{1},y_{2},\ldots ,y_{n})=P(y_{\pi _{1}},y_{\pi _{2}},\ldots ,y_{\pi _{n}}).} The following is an exchangeable, but not independent and identical (iid), example: Consider an urn with a red ball and a blue ball inside, with probability 1 2 {\displaystyle {\frac {1}{2}}} of drawing either. Balls are drawn without replacement, i.e. after one ball is drawn from the n {\displaystyle n} balls, there will be n − 1 {\displaystyle n-1} remaining balls left for the next draw. Let Y i = { 1 , if the i th ball is red , 0 , otherwise . {\displaystyle {\text{Let }}Y_{i}={\begin{cases}1,&{\text{if the }}i{\text{th ball is red}},\\0,&{\text{otherwise}}.\end{cases}}} The probability of selecting a red ball in the first draw and a blue ball in the second draw is equal to the probability of selecting a blue ball on the first draw and a red on the second, both of which are 1/2: P ( y 1 = 1 , y 2 = 0 ) = P ( y 1 = 0 , y 2 = 1 ) = 1 2 {\displaystyle P(y_{1}=1,y_{2}=0)=P(y_{1}=0,y_{2}=1)={\frac {1}{2}}} . This makes y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} exchangeable. But the probability of selecting a red ball on the second draw given that the red ball has already been selected in the first is 0. This is not equal to the probability that the red ball is selected in the second draw, which is 1/2: P ( y 2 = 1 ∣ y 1 = 1 ) = 0 ≠ P ( y 2 = 1 ) = 1 2 {\displaystyle P(y_{2}=1\mid y_{1}=1)=0\neq P(y_{2}=1)={\frac {1}{2}}} . Thus, y 1 {\displaystyle y_{1}} and y 2 {\displaystyle y_{2}} are not independent. If x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are independent and identically distributed, then they are exchangeable, but the converse is not necessarily true. === Infinite exchangeability === Infinite exchangeability is the property that every finite subset of an infinite sequence y 1 {\displaystyle y_{1}} , y 2 , … {\displaystyle y_{2},\ldots } is exchangeable. For any n, the sequence y 1 , y 2 , … , y n {\displaystyle y_{1},y_{2},\ldots ,y_{n}} is exchangeable. == Hierarchical models == === Components === Bayesian hierarchical modeling makes use of two important concepts in deriving the posterior distribution, namely: Hyperparameters: parameters of the prior distribution Hyperpriors: distributions of Hyperparameters Suppose a random variable Y follows a normal distribution with parameter θ {\displaystyle \theta } as the mean and 1 as the variance, that is Y ∣ θ ∼ N ( θ , 1 ) {\displaystyle Y\mid \theta \sim N(\theta ,1)} . The tilde relation ∼ {\displaystyle \sim } can be read as "has the distribution of" or "is distributed as". Suppose also that the parameter θ {\displaystyle \theta } has a distribution given by a normal distribution with mean μ {\displaystyle \mu } and variance 1, i.e. θ ∣ μ ∼ N ( μ , 1 ) {\displaystyle \theta \mid \mu \sim N(\mu ,1)} . Furthermore, μ {\displaystyle \mu } follows another distribution given, for example, by the standard normal distribution, N ( 0 , 1 ) {\displaystyle {\text{N}}(0,1)} . The parameter μ {\dis
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Almeida–Pineda recurrent backpropagation

Almeida–Pineda recurrent backpropagation is an extension to the backpropagation algorithm that is applicable to recurrent neural networks. It is a type of supervised learning. It was described somewhat cryptically in Richard Feynman's senior thesis, and rediscovered independently in the context of artificial neural networks by both Fernando Pineda and Luis B. Almeida. A recurrent neural network for this algorithm consists of some input units, some output units and eventually some hidden units. For a given set of (input, target) states, the network is trained to settle into a stable activation state with the output units in the target state, based on a given input state clamped on the input units.
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Application permissions

Permissions are a means of controlling and regulating access to specific system- and device-level functions by software. Typically, types of permissions cover functions that may have privacy implications, such as the ability to access a device's hardware features (including the camera and microphone), and personal data (such as storage devices, contacts lists, and the user's present geographical location). Permissions are typically declared in an application's manifest, and certain permissions must be specifically granted at runtime by the user—who may revoke the permission at any time. Permission systems are common on mobile operating systems, where permissions needed by specific apps must be disclosed via the platform's app store. == Mobile devices == On mobile operating systems for smartphones and tablets, typical types of permissions regulate: Access to storage and personal information, such as contacts, calendar appointments, etc. Location tracking. Access to the device's internal camera and/or microphone. Access to biometric sensors, including fingerprint readers and other health sensors.. Internet access. Access to communications interfaces (including their hardware identifiers and signal strength where applicable, and requests to enable them), such as Bluetooth, Wi-Fi, NFC, and others. Making and receiving phone calls. Sending and reading text messages The ability to perform in-app purchases. The ability to "overlay" themselves within other apps. Installing, deleting and otherwise managing applications. Authentication tokens (e.g., OAuth tokens) from web services stored in system storage for sharing between apps. Prior to Android 6.0 "Marshmallow", permissions were automatically granted to apps at runtime, and they were presented upon installation in Google Play Store. Since Marshmallow, certain permissions now require the app to request permission at runtime by the user. These permissions may also be revoked at any time via Android's settings menu. Usage of permissions on Android are sometimes abused by app developers to gather personal information and deliver advertising; in particular, apps for using a phone's camera flash as a flashlight (which have grown largely redundant due to the integration of such functionality at the system level on later versions of Android) have been known to require a large array of unnecessary permissions beyond what is actually needed for the stated functionality. iOS imposes a similar requirement for permissions to be granted at runtime, with particular controls offered for enabling of Bluetooth, Wi-Fi, and location tracking. == WebPermissions == WebPermissions is a permission system for web browsers. When a web application needs some data behind permission, it must request it first. When it does it, a user sees a window asking him to make a choice. The choice is remembered, but can be cleared lately. Currently the following resources are controlled: geolocation desktop notifications service workers sensors audio capturing devices, like sound cards, and their model names and characteristics video capturing devices, like cameras, and their identifiers and characteristics == Analysis == The permission-based access control model assigns access privileges for certain data objects to application. This is a derivative of the discretionary access control model. The access permissions are usually granted in the context of a specific user on a specific device. Permissions are granted permanently with few automatic restrictions. In some cases permissions are implemented in 'all-or-nothing' approach: a user either has to grant all the required permissions to access the application or the user can not access the application. There is still a lack of transparency when the permission is used by a program or application to access the data protected by the permission access control mechanism. Even if a user can revoke a permission, the app can blackmail a user by refusing to operate, for example by just crashing or asking user to grant the permission again in order to access the application. The permission mechanism has been widely criticized by researchers for several reasons, including; Intransparency of personal data extraction and surveillance, including the creation of a false sense of security; End-user fatigue of micro-managing access permissions leading to a fatalistic acceptance of surveillance and intransparency; Massive data extraction and personal surveillance carried out once the permissions are granted. Some apps, such as XPrivacy and Mockdroid spoof data in order to act as a measure for privacy. Further transparency methods include longitudinal behavioural profiling and multiple-source privacy analysis of app data access.
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Probit model

In statistics, a probit model is a type of regression where the dependent variable can take only two values, for example married or not married. The word is a portmanteau, coming from probability + unit. The purpose of the model is to estimate the probability that an observation with particular characteristics will fall into a specific one of the categories; moreover, classifying observations based on their predicted probabilities is a type of binary classification model. A probit model is a popular specification for a binary response model. As such it treats the same set of problems as does logistic regression using similar techniques. When viewed in the generalized linear model framework, the probit model employs a probit link function. It is most often estimated using the maximum likelihood procedure, such an estimation being called a probit regression. == Conceptual framework == Suppose a response variable Y is binary, that is it can have only two possible outcomes which we will denote as 1 and 0. For example, Y may represent presence/absence of a certain condition, success/failure of some device, answer yes/no on a survey, etc. We also have a vector of regressors X, which are assumed to influence the outcome Y. Specifically, we assume that the model takes the form P ( Y = 1 ∣ X ) = Φ ( X T β ) , {\displaystyle P(Y=1\mid X)=\Phi (X^{\operatorname {T} }\beta ),} where P is the probability and Φ {\displaystyle \Phi } is the cumulative distribution function (CDF) of the standard normal distribution. The parameters β are typically estimated by maximum likelihood. It is possible to motivate the probit model as a latent variable model. Suppose there exists an auxiliary random variable Y ∗ = X T β + ε , {\displaystyle Y^{\ast }=X^{T}\beta +\varepsilon ,} where ε ~ N(0, 1). Then Y can be viewed as an indicator for whether this latent variable is positive: Y = { 1 Y ∗ > 0 0 otherwise } = { 1 X T β + ε > 0 0 otherwise } {\displaystyle Y=\left.{\begin{cases}1&Y^{}>0\\0&{\text{otherwise}}\end{cases}}\right\}=\left.{\begin{cases}1&X^{\operatorname {T} }\beta +\varepsilon >0\\0&{\text{otherwise}}\end{cases}}\right\}} The use of the standard normal distribution causes no loss of generality compared with the use of a normal distribution with an arbitrary mean and standard deviation, because adding a fixed amount to the mean can be compensated by subtracting the same amount from the intercept, and multiplying the standard deviation by a fixed amount can be compensated by multiplying the weights by the same amount. To see that the two models are equivalent, note that P ( Y = 1 ∣ X ) = P ( Y ∗ > 0 ) = P ( X T β + ε > 0 ) = P ( ε > − X T β ) = P ( ε < X T β ) by symmetry of the normal distribution = Φ ( X T β ) {\displaystyle {\begin{aligned}P(Y=1\mid X)&=P(Y^{\ast }>0)\\&=P(X^{\operatorname {T} }\beta +\varepsilon >0)\\&=P(\varepsilon >-X^{\operatorname {T} }\beta )\\&=P(\varepsilon 0 {\displaystyle t,\lim _{n\rightarrow \infty }n_{t}/n=c_{t}>0} . Denote p ^ t = r t / n t {\displaystyle {\hat {p}}_{t}=r_{t}/n_{t}} σ ^ t 2 = 1 n t p ^ t ( 1 − p ^ t ) φ 2 ( Φ − 1 ( p ^ t ) ) {\displaystyle {\hat {\sigma }}_{t}^{2}={\frac {1}{n_{t}}}{\frac {{\hat {p}}_{t}(1-{\hat {p}}_{t})}{\varphi ^{2}{\big (}\Phi ^{-1}({\hat {p}}_{t}){\big )}}}} Then Berkson's minimum chi-square estimator is a generalized least squares estimator in a regression of Φ − 1 ( p ^ t ) {\displaystyle \Phi ^{-1}({\hat {p}}_{t})} on x ( t ) {\displaystyle x_{(t)}} with weights σ ^ t − 2 {\displaystyle {\hat {\sigma }}_{t}^{-2}} : β ^ = ( ∑ t = 1 T σ ^ t − 2 x ( t ) x ( t ) T ) − 1 ∑ t = 1 T σ ^ t − 2 x ( t ) Φ − 1 ( p ^ t ) {\displaystyle {\hat {\beta }}={\Bigg (}\sum _{t=1}^{T}{\hat {\sigma }}_{t}^{-2}x_{(t)}x_{(t)}^{\operatorname {T} }{\Bigg )}^{-1}\sum _{t=1}^{T}{\hat {\sigma }}_{t}^{-2}x_{(t)}\Phi ^{-1}({\hat {p}}_{t})} It can be shown that this estimator is consistent (as n→∞ and T fixed), asymptotically normal and efficient. Its advantage is the presence of a closed-form formula for the estimator. However, it is only meaningful to carry out this analysis when individual observations are not available, only their aggregated counts r t {\displaystyle r_{t}} , n t {\disp
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Multiple discriminant analysis

Multiple Discriminant Analysis (MDA) is a multivariate dimensionality reduction technique. It has been used to predict signals as diverse as neural memory traces and corporate failure. MDA is not directly used to perform classification. It merely supports classification by yielding a compressed signal amenable to classification. The method described in Duda et al. (2001) §3.8.3 projects the multivariate signal down to an M−1 dimensional space where M is the number of categories. MDA is useful because most classifiers are strongly affected by the curse of dimensionality. In other words, when signals are represented in very-high-dimensional spaces, the classifier's performance is catastrophically impaired by the overfitting problem. This problem is reduced by compressing the signal down to a lower-dimensional space as MDA does. MDA has been used to reveal neural codes.
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