Hierarchical Risk Parity (HRP) is an advanced investment portfolio optimization framework developed in 2016 by Marcos López de Prado at Guggenheim Partners and Cornell University. HRP is a probabilistic graph-based alternative to the prevailing mean-variance optimization (MVO) framework developed by Harry Markowitz in 1952, and for which he received the Nobel Prize in economic sciences. HRP algorithms apply discrete mathematics and machine learning techniques to create diversified and robust investment portfolios that outperform MVO methods out-of-sample. HRP aims to address the limitations of traditional portfolio construction methods, particularly when dealing with highly correlated assets. Following its publication, HRP has been implemented in numerous open-source libraries, and received multiple extensions. == Key features == HRP portfolios have been proposed as a robust alternative to traditional quadratic optimization methods, including the Critical Line Algorithm (CLA) of Markowitz. HRP addresses three central issues commonly associated with quadratic optimizers: numerical instability, excessive concentration in a small number of assets, and poor out-of-sample performance. HRP leverages techniques from graph theory and machine learning to construct diversified portfolios using only the information embedded in the covariance matrix. Unlike quadratic programming methods, HRP does not require the covariance matrix to be invertible. Consequently, HRP remains applicable even in cases where the covariance matrix is ill-conditioned or singular—conditions under which standard optimizers fail. Monte Carlo simulations indicate that HRP achieves lower out-of-sample variance than CLA, despite the fact that minimizing variance is the explicit optimization objective of CLA. Furthermore, HRP portfolios exhibit lower realized risk compared to those generated by traditional risk parity methodologies. Empirical backtests have demonstrated that HRP would have historically outperformed conventional portfolio construction techniques. Algorithms within the HRP framework are characterized by the following features: Machine Learning Approach: HRP employs hierarchical clustering, a machine learning technique, to group similar assets based on their correlations. This allows the algorithm to identify the underlying hierarchical structure of the portfolio, and avoid that errors spread through the entire network. Risk-Based Allocation: The algorithm allocates capital based on risk, ensuring that assets only compete with similar assets for representation in the portfolio. This approach leads to better diversification across different risk sources, while avoiding the instability associated with noisy returns estimates. Covariance Matrix Handling: Unlike traditional methods like Mean-Variance Optimization, HRP does not require inverting the covariance matrix. This makes it more stable and applicable to portfolios with a large number of assets, particularly when the covariance matrix's condition number is high. == The problem: Markowitz's Curse == Portfolio construction is perhaps the most recurrent financial problem. On a daily basis, investment managers must build portfolios that incorporate their views and forecasts on risks and returns. Despite the theoretical elegance of Markowitz's mean-variance framework, its practical implementation is hindered by several limitations that undermine the reliability of solutions derived from the Critical Line Algorithm (CLA). A principal concern is the high sensitivity of optimal portfolios to small perturbations in expected returns: even minor forecasting errors can result in significantly different allocations (Michaud, 1998). Given the inherent difficulty of producing accurate return forecasts, numerous researchers have advocated for approaches that forgo expected returns entirely and instead rely solely on the covariance structure of asset returns. This has given rise to risk-based allocation methods, among which risk parity is a widely cited example (Jurczenko, 2015). While eliminating return forecasts mitigates some instability, it does not eliminate it. Quadratic programming techniques employed in portfolio optimization require the inversion of a positive-definite covariance matrix, meaning all eigenvalues must be strictly positive. When the matrix is numerically ill-conditioned—that is, when the ratio of its largest to smallest eigenvalue (its condition number) is large—matrix inversion becomes unreliable and prone to significant numerical errors (Bailey and López de Prado, 2012). The condition number of a covariance, correlation, or any symmetric (and thus diagonalizable) matrix is defined as the absolute value of the ratio between its largest and smallest eigenvalues in modulus. The figure on the right presents the sorted eigenvalues of several correlation matrices; the condition number is represented by the ratio of the first to last eigenvalues in each sequence. A diagonal correlation matrix, which is equal to its own inverse, exhibits the minimum possible condition number. As the number of correlated (or multicollinear) assets in a portfolio increases, the condition number rises. At high levels, this leads to severe numerical instability, whereby slight modifications in any matrix entry may result in drastically different inverses. This phenomenon, often referred to as Markowitz’s curse, encapsulates the paradox wherein increased correlation among assets heightens the theoretical need for diversification, yet simultaneously increases the likelihood of unstable optimization outcomes. Consequently, the potential benefits of diversification are frequently overshadowed by estimation errors. These problems are exacerbated as the dimensionality of the covariance matrix increases. The estimation of each covariance term consumes degrees of freedom, and in general, a minimum of 1 2 N ( N + 1 ) {\displaystyle {\frac {1}{2}}N(N+1)} independent and identically distributed (IID) observations is required to estimate a non-singular covariance matrix of dimension N {\displaystyle N} . For example, constructing an invertible covariance matrix of dimension 50 necessitates at least five years of daily IID observations. However, empirical evidence suggests that the correlation structure of financial assets is highly unstable over such extended periods. These difficulties are highlighted by the observation that even naïve allocation strategies—such as equally weighted portfolios—have frequently outperformed both mean-variance and risk-based optimizations in out-of-sample tests (De Miguel et al., 2009). == The solution: Hierarchical Risk Parity == The HRP algorithm addresses Markowitz's curse in three steps: Hierarchical Clustering: Assets are grouped into clusters based on their correlations, forming a hierarchical tree structure. Quasi-Diagonalization: The correlation matrix is reordered based on the clustering results, revealing a block diagonal structure. Recursive Bisection: Weights are assigned to assets through a top-down approach, splitting the portfolio into smaller sub-portfolios and allocating capital based on inverse variance. === Step 1: Hierarchical clustering === Given a T × N {\displaystyle T\times N} matrix of asset returns X {\displaystyle X} , where each column represents a time series of returns for one of N {\displaystyle N} assets over T {\displaystyle T} time periods, a hierarchical clustering process can be used to construct a tree-based representation of asset relationships. First, we compute the N × N {\displaystyle N\times N} correlation matrix ρ = ρ i , j i , j = 1 . . . N {\displaystyle \rho ={\rho _{i,j}}\;{i,j=1\;...\;N}} , where ρ i , j = c o r r ( X i , X j ) {\displaystyle \rho _{i,j}=\mathrm {corr} (X_{i},X_{j})} . From this, a pairwise distance matrix D = d i , j {\displaystyle D={d_{i,j}}} is defined using the transformation: d i , j = 1 2 ( 1 − ρ i , j ) {\displaystyle d_{i,j}={\sqrt {{\frac {1}{2}}(1-\rho _{i,j})}}} This distance function defines a proper metric space, satisfying non-negativity, identity of indiscernibles, symmetry, and the triangle inequality. Next, a secondary distance matrix D ~ = d ~ i , j {\displaystyle {\tilde {D}}={{\tilde {d}}_{i,j}}} is computed, where each entry measures the Euclidean distance between the distance profiles of two assets: d ~ i , j = ∑ n = 1 N ( d n , i − d n , j ) 2 {\displaystyle {\tilde {d}}_{i,j}={\sqrt {\sum _{n=1}^{N}(d_{n,i}-d_{n,j})^{2}}}} While d i , j {\displaystyle d_{i,j}} reflects correlation-based proximity between two assets, d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} quantifies dissimilarity across the entire system, as it depends on all pairwise distances. Hierarchical clustering proceeds by identifying the pair ( i , j ) {\displaystyle (i,j)} with the smallest value of d ~ i , j {\displaystyle {\tilde {d}}_{i,j}} (for i ≠ j {\displaystyle i\neq j} ), and forming a new cluster u [ 1 ] = ( i , j ) {\displaystyle u[1]=(i,j)} .
Apache Kudu
Apache Kudu is a free and open source column-oriented data store of the Apache Hadoop ecosystem. It is compatible with most of the data processing frameworks in the Hadoop environment. It provides completeness to Hadoop's storage layer to enable fast analytics on fast data. The open source project to build Apache Kudu began as internal project at Cloudera. The first version Apache Kudu 1.0 was released 19 September 2016. == Comparison with other storage engines == Kudu was designed and optimized for OLAP workloads. Like HBase, it is a real-time store that supports key-indexed record lookup and mutation. Kudu differs from HBase since Kudu's datamodel is a more traditional relational model, while HBase is schemaless. Kudu's "on-disk representation is truly columnar and follows an entirely different storage design than HBase/Bigtable".
European Conference on Computer Vision
The European Conference on Computer Vision (ECCV) is a biennial research conference with the proceedings published by Springer Science+Business Media. Similar to ICCV in scope and quality, it is held those years which ICCV is not. It is considered to be one of the top conferences in computer vision, alongside CVPR and ICCV, with an 'A' rating from the Australian Ranking of ICT Conferences and an 'A1' rating from the Brazilian ministry of education. The acceptance rate for ECCV 2010 was 24.4% for posters and 3.3% for oral presentations. Like other top computer vision conferences, ECCV has tutorial talks, technical sessions, and poster sessions. The conference is usually spread over five to six days with the main technical program occupying three days in the middle, and tutorial and workshops, focused on specific topics, being held in the beginning and at the end. The ECCV presents the Koenderink Prize annually to recognize fundamental contributions in computer vision. == Location == The conference is usually held in autumn in Europe.
FastICA
FastICA is an efficient and popular algorithm for independent component analysis invented by Aapo Hyvärinen at Helsinki University of Technology. Like most ICA algorithms, FastICA seeks an orthogonal rotation of prewhitened data, through a fixed-point iteration scheme, that maximizes a measure of non-Gaussianity of the rotated components. Non-gaussianity serves as a proxy for statistical independence, which is a very strong condition and requires infinite data to verify. FastICA can also be alternatively derived as an approximative Newton iteration. == Algorithm == === Prewhitening the data === Let the X := ( x i j ) ∈ R N × M {\displaystyle \mathbf {X} :=(x_{ij})\in \mathbb {R} ^{N\times M}} denote the input data matrix, M {\displaystyle M} the number of columns corresponding with the number of samples of mixed signals and N {\displaystyle N} the number of rows corresponding with the number of independent source signals. The input data matrix X {\displaystyle \mathbf {X} } must be prewhitened, or centered and whitened, before applying the FastICA algorithm to it. Centering the data entails demeaning each component of the input data X {\displaystyle \mathbf {X} } , that is, for each i = 1 , … , N {\displaystyle i=1,\ldots ,N} and j = 1 , … , M {\displaystyle j=1,\ldots ,M} . After centering, each row of X {\displaystyle \mathbf {X} } has an expected value of 0 {\displaystyle 0} . Whitening the data requires a linear transformation L : R N × M → R N × M {\displaystyle \mathbf {L} :\mathbb {R} ^{N\times M}\to \mathbb {R} ^{N\times M}} of the centered data so that the components of L ( X ) {\displaystyle \mathbf {L} (\mathbf {X} )} are uncorrelated and have variance one. More precisely, if X {\displaystyle \mathbf {X} } is a centered data matrix, the covariance of L x := L ( X ) {\displaystyle \mathbf {L} _{\mathbf {x} }:=\mathbf {L} (\mathbf {X} )} is the ( N × N ) {\displaystyle (N\times N)} -dimensional identity matrix, that is, A common method for whitening is by performing an eigenvalue decomposition on the covariance matrix of the centered data X {\displaystyle \mathbf {X} } , E { X X T } = E D E T {\displaystyle E\left\{\mathbf {X} \mathbf {X} ^{T}\right\}=\mathbf {E} \mathbf {D} \mathbf {E} ^{T}} , where E {\displaystyle \mathbf {E} } is the matrix of eigenvectors and D {\displaystyle \mathbf {D} } is the diagonal matrix of eigenvalues. The whitened data matrix is defined thus by === Single component extraction === The iterative algorithm finds the direction for the weight vector w ∈ R N {\displaystyle \mathbf {w} \in \mathbb {R} ^{N}} that maximizes a measure of non-Gaussianity of the projection w T X {\displaystyle \mathbf {w} ^{T}\mathbf {X} } , with X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} denoting a prewhitened data matrix as described above. Note that w {\displaystyle \mathbf {w} } is a column vector. To measure non-Gaussianity, FastICA relies on a nonquadratic nonlinear function f ( u ) {\displaystyle f(u)} , its first derivative g ( u ) {\displaystyle g(u)} , and its second derivative g ′ ( u ) {\displaystyle g^{\prime }(u)} . Hyvärinen states that the functions are useful for general purposes, while may be highly robust. The steps for extracting the weight vector w {\displaystyle \mathbf {w} } for single component in FastICA are the following: Randomize the initial weight vector w {\displaystyle \mathbf {w} } Let w + ← E { X g ( w T X ) T } − E { g ′ ( w T X ) } w {\displaystyle \mathbf {w} ^{+}\leftarrow E\left\{\mathbf {X} g(\mathbf {w} ^{T}\mathbf {X} )^{T}\right\}-E\left\{g'(\mathbf {w} ^{T}\mathbf {X} )\right\}\mathbf {w} } , where E { . . . } {\displaystyle E\left\{...\right\}} means averaging over all column-vectors of matrix X {\displaystyle \mathbf {X} } Let w ← w + / ‖ w + ‖ {\displaystyle \mathbf {w} \leftarrow \mathbf {w} ^{+}/\|\mathbf {w} ^{+}\|} If not converged, go back to 2 === Multiple component extraction === The single unit iterative algorithm estimates only one weight vector which extracts a single component. Estimating additional components that are mutually "independent" requires repeating the algorithm to obtain linearly independent projection vectors - note that the notion of independence here refers to maximizing non-Gaussianity in the estimated components. Hyvärinen provides several ways of extracting multiple components with the simplest being the following. Here, 1 M {\displaystyle \mathbf {1_{M}} } is a column vector of 1's of dimension M {\displaystyle M} . Algorithm FastICA Input: C {\displaystyle C} Number of desired components Input: X ∈ R N × M {\displaystyle \mathbf {X} \in \mathbb {R} ^{N\times M}} Prewhitened matrix, where each column represents an N {\displaystyle N} -dimensional sample, where C <= N {\displaystyle C<=N} Output: W ∈ R N × C {\displaystyle \mathbf {W} \in \mathbb {R} ^{N\times C}} Un-mixing matrix where each column projects X {\displaystyle \mathbf {X} } onto independent component. Output: S ∈ R C × M {\displaystyle \mathbf {S} \in \mathbb {R} ^{C\times M}} Independent components matrix, with M {\displaystyle M} columns representing a sample with C {\displaystyle C} dimensions. for p in 1 to C: w p ← {\displaystyle \mathbf {w_{p}} \leftarrow } Random vector of length N while w p {\displaystyle \mathbf {w_{p}} } changes w p ← 1 M X g ( w p T X ) T − 1 M g ′ ( w p T X ) 1 M w p {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {1}{M}}\mathbf {X} g(\mathbf {w_{p}} ^{T}\mathbf {X} )^{T}-{\frac {1}{M}}g'(\mathbf {w_{p}} ^{T}\mathbf {X} )\mathbf {1_{M}} \mathbf {w_{p}} } w p ← w p − ∑ j = 1 p − 1 ( w p T w j ) w j {\displaystyle \mathbf {w_{p}} \leftarrow \mathbf {w_{p}} -\sum _{j=1}^{p-1}(\mathbf {w_{p}} ^{T}\mathbf {w_{j}} )\mathbf {w_{j}} } w p ← w p ‖ w p ‖ {\displaystyle \mathbf {w_{p}} \leftarrow {\frac {\mathbf {w_{p}} }{\|\mathbf {w_{p}} \|}}} output W ← [ w 1 , … , w C ] {\displaystyle \mathbf {W} \leftarrow {\begin{bmatrix}\mathbf {w_{1}} ,\dots ,\mathbf {w_{C}} \end{bmatrix}}} output S ← W T X {\displaystyle \mathbf {S} \leftarrow \mathbf {W^{T}} \mathbf {X} }
Sharpness aware minimization
Sharpness Aware Minimization (SAM) is an optimization algorithm used in machine learning that aims to improve model generalization. The method seeks to find model parameters that are located in regions of the loss landscape with uniformly low loss values, rather than parameters that only achieve a minimal loss value at a single point. This approach is described as finding "flat" minima instead of "sharp" ones. The rationale is that models trained this way are less sensitive to variations between training and test data, which can lead to better performance on unseen data. The algorithm was introduced in a 2020 paper by a team of researchers including Pierre Foret, Ariel Kleiner, Hossein Mobahi, and Behnam Neyshabur. == Underlying Principle == SAM modifies the standard training objective by minimizing a "sharpness-aware" loss. This is formulated as a minimax problem where the inner objective seeks to find the highest loss value in the immediate neighborhood of the current model weights, and the outer objective minimizes this value: min w max ‖ ϵ ‖ p ≤ ρ L train ( w + ϵ ) + λ ‖ w ‖ 2 2 {\displaystyle \min _{w}\max _{\|\epsilon \|_{p}\leq \rho }L_{\text{train}}(w+\epsilon )+\lambda \|w\|_{2}^{2}} In this formulation: w {\displaystyle w} represents the model's parameters (weights). L train {\displaystyle L_{\text{train}}} is the loss calculated on the training data. ϵ {\displaystyle \epsilon } is a perturbation applied to the weights. ρ {\displaystyle \rho } is a hyperparameter that defines the radius of the neighborhood (an L p {\displaystyle L_{p}} ball) to search for the highest loss. An optional L2 regularization term, scaled by λ {\displaystyle \lambda } , can be included. A direct solution to the inner maximization problem is computationally expensive. SAM approximates it by taking a single gradient ascent step to find the perturbation ϵ {\displaystyle \epsilon } . This is calculated as: ϵ ( w ) = ρ ∇ L train ( w ) ‖ ∇ L train ( w ) ‖ 2 {\displaystyle \epsilon (w)=\rho {\frac {\nabla L_{\text{train}}(w)}{\|\nabla L_{\text{train}}(w)\|_{2}}}} The optimization process for each training step involves two stages. First, an "ascent step" computes a perturbed set of weights, w adv = w + ϵ ( w ) {\displaystyle w_{\text{adv}}=w+\epsilon (w)} , by moving towards the direction of the highest local loss. Second, a "descent step" updates the original weights w {\displaystyle w} using the gradient calculated at these perturbed weights, ∇ L train ( w adv ) {\displaystyle \nabla L_{\text{train}}(w_{\text{adv}})} . This update is typically performed using a standard optimizer like SGD or Adam. == Application and Performance == SAM has been applied in various machine learning contexts, primarily in computer vision. Research has shown it can improve generalization performance in models such as Convolutional Neural Networks (CNNs) and Vision Transformers (ViTs) on image datasets including ImageNet, CIFAR-10, and CIFAR-100. The algorithm has also been found to be effective in training models with noisy labels, where it performs comparably to methods designed specifically for this problem. Some studies indicate that SAM and its variants can improve out-of-distribution (OOD) generalization, which is a model's ability to perform well on data from distributions not seen during training. Other areas where it has been applied include gradual domain adaptation and mitigating overfitting in scenarios with repeated exposure to training examples. == Limitations == A primary limitation of SAM is its computational cost. By requiring two gradient computations (one for the ascent and one for the descent) per optimization step, it approximately doubles the training time compared to standard optimizers. The theoretical convergence properties of SAM are still under investigation. Some research suggests that with a constant step size, SAM may not converge to a stationary point. The accuracy of the single gradient step approximation for finding the worst-case perturbation may also decrease during the training process. The effectiveness of SAM can also be domain-dependent. While it has shown benefits for computer vision tasks, its impact on other areas, such as GPT-style language models where each training example is seen only once, has been reported as limited in some studies. Furthermore, while SAM seeks flat minima, some research suggests that not all flat minima necessarily lead to good generalization. The algorithm also introduces the neighborhood size ρ {\displaystyle \rho } as a new hyperparameter, which requires tuning. == Research, Variants, and Enhancements == Active research on SAM focuses on reducing its computational overhead and improving its performance. Several variants have been proposed to make the algorithm more efficient. These include methods that attempt to parallelize the two gradient computations, apply the perturbation to only a subset of parameters, or reduce the number of computation steps required. Other approaches use historical gradient information or apply SAM steps intermittently to lower the computational burden. To improve performance and robustness, variants have been developed that adapt the neighborhood size based on model parameter scales (Adaptive SAM or ASAM) or incorporate information about the curvature of the loss landscape (Curvature Regularized SAM or CR-SAM). Other research explores refining the perturbation step by focusing on specific components of the gradient or combining SAM with techniques like random smoothing. Theoretical work continues to analyze the algorithm's behavior, including its implicit bias towards flatter minima and the development of broader frameworks for sharpness-aware optimization that use different measures of sharpness.
Graphics software
In computer graphics, graphics software refers to a program or collection of programs that enable a person to manipulate images or models visually on a computer. Computer graphics can be classified into two distinct categories: raster graphics and vector graphics, with further 2D and 3D variants. Many graphics programs focus exclusively on either vector or raster graphics, but there are a few that operate on both. It is simple to convert from vector graphics to raster graphics, but going the other way is harder. Some software attempts to do this. In addition to static graphics, there are animation and video editing software. Different types of software are often designed to edit different types of graphics such as video, photos, and vector-based drawings. The exact sources of graphics may vary for different tasks, but most can read and write files. Most graphics programs have the ability to import and export one or more graphics file formats, including those formats written for a particular computer graphics program. Such programs include, but are not limited to: GIMP, Adobe Photoshop, CorelDRAW, Microsoft Publisher, Picasa, etc. The use of a swatch is a palette of active colours that are selected and rearranged by the preference of the user. A swatch may be used in a program or be part of the universal palette on an operating system. It is used to change the colour of a text or image and in video editing. Vector graphics animation can be described as a series of mathematical transformations that are applied in sequence to one or more shapes in a scene. Raster graphics animation works in a similar fashion to film-based animation, where a series of still images produces the illusion of continuous movement. == History == SuperPaint was one of the earliest graphics software applications, first conceptualized in 1972 and achieving its first stable image in 1973 Fauve Matisse (later Macromedia xRes) was a pioneering program of the early 1990s, notably introducing layers in customer software. Currently Adobe Photoshop is one of the most used and best-known graphics programs in the Americas, having created more custom hardware solutions in the early 1990s, but was initially subject to various litigation. GIMP is a popular open-source alternative to Adobe Photoshop.
Perceptron
In machine learning, the perceptron is an algorithm for supervised learning of binary classifiers. A binary classifier is a function that can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier, i.e. a classification algorithm that makes its predictions based on a linear predictor function combining a set of weights with the feature vector. == History == The artificial neuron and artificial neural network were invented in 1943 by Warren McCulloch and Walter Pitts in their seminal paper "A Logical Calculus of the Ideas Immanent in Nervous Activity". In 1957, Frank Rosenblatt was at the Cornell Aeronautical Laboratory. He simulated the perceptron on an IBM 704. Later, he obtained funding by the Information Systems Branch of the United States Office of Naval Research and the Rome Air Development Center, to build a custom-made computer, the Mark I Perceptron. It was first publicly demonstrated on 23 June 1960. The machine was "part of a previously secret four-year NPIC [the US' National Photographic Interpretation Center] effort from 1963 through 1966 to develop this algorithm into a useful tool for photo-interpreters". Rosenblatt described the details of the perceptron in a 1958 paper. His organization of a perceptron is constructed of three kinds of cells ("units"): S, A, R, which stand for "sensory", "association" and "response". He presented at the first international symposium on AI, Mechanisation of Thought Processes, which took place in 1958 November. Rosenblatt's project was funded under Contract Nonr-401(40) "Cognitive Systems Research Program", which lasted from 1959 to 1970, and Contract Nonr-2381(00) "Project PARA" ("PARA" means "Perceiving and Recognition Automata"), which lasted from 1957 to 1963. In 1959, the Institute for Defense Analysis awarded his group a $10,000 contract. By September 1961, the ONR awarded further $153,000 worth of contracts, with $108,000 committed for 1962. The ONR research manager, Marvin Denicoff, stated that ONR, instead of ARPA, funded the Perceptron project, because the project was unlikely to produce technological results in the near or medium term. Funding from ARPA go up to the order of millions dollars, while from ONR are on the order of 10,000 dollars. Meanwhile, the head of IPTO at ARPA, J.C.R. Licklider, was interested in 'self-organizing', 'adaptive' and other biologically-inspired methods in the 1950s; but by the mid-1960s he was openly critical of these, including the perceptron. Instead he strongly favored the logical AI approach of Simon and Newell. === Mark I Perceptron machine === The perceptron was intended to be a machine, rather than a program, and while its first implementation was in software for the IBM 704, it was subsequently implemented in custom-built hardware as the Mark I Perceptron with the project name "Project PARA", designed for image recognition. The machine is currently in Smithsonian National Museum of American History. The Mark I Perceptron had three layers. One version was implemented as follows: An array of 400 photocells arranged in a 20x20 grid, named "sensory units" (S-units), or "input retina". Each S-unit can connect to up to 40 A-units. A hidden layer of 512 perceptrons, named "association units" (A-units). An output layer of eight perceptrons, named "response units" (R-units). Rosenblatt called this three-layered perceptron network the alpha-perceptron, to distinguish it from other perceptron models he experimented with. The S-units are connected to the A-units randomly (according to a table of random numbers) via a plugboard (see photo), to "eliminate any particular intentional bias in the perceptron". The connection weights are fixed, not learned. Rosenblatt was adamant about the random connections, as he believed the retina was randomly connected to the visual cortex, and he wanted his perceptron machine to resemble human visual perception. The A-units are connected to the R-units, with adjustable weights encoded in potentiometers, and weight updates during learning were performed by electric motors.The hardware details are in an operators' manual. In a 1958 press conference organized by the US Navy, Rosenblatt made statements about the perceptron that caused a heated controversy among the fledgling AI community; based on Rosenblatt's statements, The New York Times reported the perceptron to be "the embryo of an electronic computer that [the Navy] expects will be able to walk, talk, see, write, reproduce itself and be conscious of its existence." The Photo Division of Central Intelligence Agency, from 1960 to 1964, studied the use of Mark I Perceptron machine for recognizing militarily interesting silhouetted targets (such as planes and ships) in aerial photos. === Principles of Neurodynamics (1962) === Rosenblatt described his experiments with many variants of the Perceptron machine in a book Principles of Neurodynamics (1962). The book is a published version of the 1961 report. Among the variants are: "cross-coupling" (connections between units within the same layer) with possibly closed loops, "back-coupling" (connections from units in a later layer to units in a previous layer), four-layer perceptrons where the last two layers have adjustable weights (and thus a proper multilayer perceptron), incorporating time-delays to perceptron units, to allow for processing sequential data, analyzing audio (instead of images). The machine was shipped from Cornell to Smithsonian in 1967, under a government transfer administered by the Office of Naval Research. === Perceptrons (1969) === Although the perceptron initially seemed promising, it was quickly proved that perceptrons could not be trained to recognise many classes of patterns. This caused the field of neural network research to stagnate for many years, before it was recognised that a feedforward neural network with two or more layers (also called a multilayer perceptron) had greater processing power than perceptrons with one layer (also called a single-layer perceptron). Single-layer perceptrons are only capable of learning linearly separable patterns. For a classification task with some step activation function, a single node will have a single line dividing the data points forming the patterns. More nodes can create more dividing lines, but those lines must somehow be combined to form more complex classifications. A second layer of perceptrons, or even linear nodes, are sufficient to solve many otherwise non-separable problems. In 1969, a famous book entitled Perceptrons by Marvin Minsky and Seymour Papert showed that it was impossible for these classes of network to learn an XOR function. It is often incorrectly believed that they also conjectured that a similar result would hold for a multi-layer perceptron network. However, this is not true, as both Minsky and Papert already knew that multi-layer perceptrons were capable of producing an XOR function. (See the page on Perceptrons (book) for more information.) Nevertheless, the often-miscited Minsky and Papert text caused a significant decline in interest and funding of neural network research. It took ten more years until neural network research experienced a resurgence in the 1980s. This text was reprinted in 1987 as "Perceptrons - Expanded Edition" where some errors in the original text are shown and corrected. === Subsequent work === Rosenblatt continued working on perceptrons despite diminishing funding. The last attempt was Tobermory, built between 1961 and 1967, built for speech recognition. It occupied an entire room. It had 4 layers with 12,000 weights implemented by toroidal magnetic cores. By the time of its completion, simulation on digital computers had become faster than purpose-built perceptron machines. He died in a boating accident in 1971. A simulation program for neural networks was written for IBM 7090/7094, and was used to study various pattern recognition applications, such as character recognition, particle tracks in bubble-chamber photographs; phoneme, isolated word, and continuous speech recognition; speaker verification; and center-of-attention mechanisms for image processing. The kernel perceptron algorithm was already introduced in 1964 by Aizerman et al. Margin bounds guarantees were given for the Perceptron algorithm in the general non-separable case first by Freund and Schapire (1998), and more recently by Mohri and Rostamizadeh (2013) who extend previous results and give new and more favorable L1 bounds. The perceptron is a simplified model of a biological neuron. While the complexity of biological neuron models is often required to fully understand neural behavior, research suggests a perceptron-like linear model can produce some behavior seen in real neurons. The solution spaces of decision boundaries for all binary functions and learning behaviors are studied in. == Definition == In the modern sense, the perceptron is an algori