In mathematics, a sequence of positive real numbers ( s 1 , s 2 , . . . ) {\displaystyle (s_{1},s_{2},...)} is called superincreasing if every element of the sequence is greater than the sum of all previous elements in the sequence. Formally, this condition can be written as s n + 1 > ∑ j = 1 n s j {\displaystyle s_{n+1}>\sum _{j=1}^{n}s_{j}} for all n ≥ 1. == Program == The following Python source code tests a sequence of numbers to determine if it is superincreasing: This produces the following output: Sum: 0 Element: 1 Sum: 1 Element: 3 Sum: 4 Element: 6 Sum: 10 Element: 13 Sum: 23 Element: 27 Sum: 50 Element: 52 Is it a superincreasing sequence? True == Examples == (1, 3, 6, 13, 27, 52) is a superincreasing sequence, but (1, 3, 4, 9, 15, 25) is not. The series a^x for a>=2 == Properties == Multiplying a superincreasing sequence by a positive real constant keeps it superincreasing.
SUPS
In computational neuroscience, SUPS (for Synaptic Updates Per Second) or formerly CUPS (Connections Updates Per Second) is a measure of a neuronal network performance, useful in fields of neuroscience, cognitive science, artificial intelligence, and computer science. == Computing == For a processor or computer designed to simulate a neural network SUPS is measured as the product of simulated neurons N {\displaystyle N} and average connectivity c {\displaystyle c} (synapses) per neuron per second: S U P S = c × N {\displaystyle SUPS=c\times N} Depending on the type of simulation it is usually equal to the total number of synapses simulated. In an "asynchronous" dynamic simulation if a neuron spikes at υ {\displaystyle \upsilon } Hz, the average rate of synaptic updates provoked by the activity of that neuron is υ c N {\displaystyle \upsilon cN} . In a synchronous simulation with step Δ t {\displaystyle \Delta t} the number of synaptic updates per second would be c N Δ t {\displaystyle {\frac {cN}{\Delta t}}} . As Δ t {\displaystyle \Delta t} has to be chosen much smaller than the average interval between two successive afferent spikes, which implies Δ t < 1 υ N {\displaystyle \Delta t<{\frac {1}{\upsilon N}}} , giving an average of synaptic updates equal to υ c N 2 {\displaystyle \upsilon cN^{2}} . Therefore, spike-driven synaptic dynamics leads to a linear scaling of computational complexity O(N) per neuron, compared with the O(N2) in the "synchronous" case. == Records == Developed in the 1980s Adaptive Solutions' CNAPS-1064 Digital Parallel Processor chip is a full neural network (NNW). It was designed as a coprocessor to a host and has 64 sub-processors arranged in a 1D array and operating in a SIMD mode. Each sub-processor can emulate one or more neurons and multiple chips can be grouped together. At 25 MHz it is capable of 1.28 GMAC. After the presentation of the RN-100 (12 MHz) single neuron chip at Seattle 1991 Ricoh developed the multi-neuron chip RN-200. It had 16 neurons and 16 synapses per neuron. The chip has on-chip learning ability using a proprietary backdrop algorithm. It came in a 257-pin PGA encapsulation and drew 3.0 W at a maximum. It was capable of 3 GCPS (1 GCPS at 32 MHz). In 1991–97, Siemens developed the MA-16 chip, SYNAPSE-1 and SYNAPSE-3 Neurocomputer. The MA-16 was a fast matrix-matrix multiplier that can be combined to form systolic arrays. It could process 4 patterns of 16 elements each (16-bit), with 16 neuron values (16-bit) at a rate of 800 MMAC or 400 MCPS at 50 MHz. The SYNAPSE3-PC PCI card contained 2 MA-16 with a peak performance of 2560 MOPS (1.28 GMAC); 7160 MOPS (3.58 GMAC) when using three boards. In 2013, the K computer was used to simulate a neural network of 1.73 billion neurons with a total of 10.4 trillion synapses (1% of the human brain). The simulation ran for 40 minutes to simulate 1 s of brain activity at a normal activity level (4.4 on average). The simulation required 1 Petabyte of storage.
Matchbox Educable Noughts and Crosses Engine
The Matchbox Educable Noughts and Crosses Engine (sometimes called the Machine Educable Noughts and Crosses Engine or MENACE) was a mechanical computer made from 304 matchboxes designed and built by artificial intelligence researcher Donald Michie and his colleague Roger Chambers, in 1961. It was designed to play human opponents in games of noughts and crosses (tic-tac-toe) by returning a move for any given state of play and to refine its strategy through reinforcement learning. This was one of the first types of artificial intelligence. Michie and Chambers did not have immediate access to a computer; they worked around this by building the engine out of matchboxes. The matchboxes they used each represented a single possible layout of a noughts and crosses grid. When the computer first played, it would randomly choose moves based on the current layout. As it played more games, through a reinforcement loop, it disqualified strategies that led to losing games, and supplemented strategies that led to winning games. Michie held a tournament against MENACE in 1961, wherein he experimented with different openings. Following MENACE's maiden tournament against Michie, it demonstrated successful artificial intelligence in its strategy. Michie's essays on MENACE's weight initialisation and the BOXES algorithm used by MENACE became popular in the field of computer science research. Michie was honoured for his contribution to machine learning research, and was twice commissioned to program a MENACE simulation on an actual computer. == Origin == Donald Michie (1923–2007) had been on the team decrypting the German Tunny Code during World War II. Fifteen years later, he wanted to further display his mathematical and computational prowess with an early convolutional neural network. Since computer equipment was not obtainable for such uses, and Michie did not have a computer readily available, he decided to display and demonstrate artificial intelligence in a more esoteric format and constructed a functional mechanical computer out of matchboxes and beads. MENACE was constructed as the result of a bet with a computer science colleague who postulated that such a machine was impossible. Michie undertook the task of collecting and defining each matchbox as a "fun project", later turned into a demonstration tool. Michie completed his essay on MENACE in 1963, "Experiments on the mechanization of game-learning", as well as his essay on the BOXES Algorithm, written with R. A. Chambers and had built up an AI research unit in Hope Park Square, Edinburgh, Scotland. MENACE learned by playing successive matches of noughts and crosses. Each time, it would eliminate a losing strategy by the human player confiscating the beads that corresponded to each move. It reinforced winning strategies by making the moves more likely, by supplying extra beads. This was one of the earliest versions of the Reinforcement Loop, the schematic algorithm of looping the algorithm, dropping unsuccessful strategies until only the winning ones remain. This model starts as completely random, and gradually learns. == Composition == MENACE was made from 304 matchboxes glued together in an arrangement similar to a chest of drawers. Each box had a code number, which was keyed into a chart. This chart had drawings of tic-tac-toe game grids with various configurations of X, O, and empty squares, corresponding to all possible permutations a game could go through as it progressed. After removing duplicate arrangements (ones that were simply rotations or mirror images of other configurations), MENACE used 304 permutations in its chart and thus that many matchboxes. Each individual matchbox tray contained a collection of coloured beads. Each colour represented a move on a square on the game grid, and so matchboxes with arrangements where positions on the grid were already taken would not have beads for that position. Additionally, at the front of the tray were two extra pieces of card in a "V" shape, the point of the "V" pointing at the front of the matchbox. Michie and his artificial intelligence team called MENACE's algorithm "Boxes", after the apparatus used for the machine. The first stage "Boxes" operated in five phases, each setting a definition and a precedent for the rules of the algorithm in relation to the game. == Operation == MENACE played first, as O, since all matchboxes represented permutations only relevant to the "X" player. To retrieve MENACE's choice of move, the opponent or operator located the matchbox that matched the current game state, or a rotation or mirror image of it. For example, at the start of a game, this would be the matchbox for an empty grid. The tray would be removed and lightly shaken so as to move the beads around. Then, the bead that had rolled into the point of the "V" shape at the front of the tray was the move MENACE had chosen to make. Its colour was then used as the position to play on, and, after accounting for any rotations or flips needed based on the chosen matchbox configuration's relation to the current grid, the O would be placed on that square. Then the player performed their move, the new state was located, a new move selected, and so on, until the game was finished. When the game had finished, the human player observed the game's outcome. As a game was played, each matchbox that was used for MENACE's turn had its tray returned to it ajar, and the bead used kept aside, so that MENACE's choice of moves and the game states they belonged to were recorded. Michie described his reinforcement system with "reward" and "punishment". Once the game was finished, if MENACE had won, it would then receive a "reward" for its victory. The removed beads showed the sequence of the winning moves. These were returned to their respective trays, easily identifiable since they were slightly open, as well as three bonus beads of the same colour. In this way, in future games MENACE would become more likely to repeat those winning moves, reinforcing winning strategies. If it lost, the removed beads were not returned, "punishing" MENACE, and meaning that in future it would be less likely, and eventually incapable if that colour of bead became absent, to repeat the moves that cause a loss. If the game was a draw, one additional bead was added to each box. == Results in practice == === Optimal strategy === Noughts and crosses has a well-known optimal strategy. A player must place their symbol in a way that blocks the other player from achieving any rows while simultaneously making a row themself. However, if both players use this strategy, the game always ends in a draw. If the human player is familiar with the optimal strategy, and MENACE can quickly learn it, then the games will eventually only end in draws. The likelihood of the computer winning increases quickly when the computer plays against a random-playing opponent. When playing against a player using optimal strategy, the odds of a draw grow to 100%. In Donald Michie's official tournament against MENACE in 1961 he used optimal strategy, and he and the computer began to draw consistently after twenty games. Michie's tournament had the following milestones: Michie began by consistently opening with "Variant 0", the middle square. At 15 games, MENACE abandoned all non-corner openings. At just over 20, Michie switched to consistently using "Variant 1", the bottom-right square. At 60, he returned to Variant 0. As he neared 80 games, he moved to "Variant 2", the top-middle. At 110, he switched to "Variant 3", the top right. At 135, he switched to "Variant 4", middle-right. At 190, he returned to Variant 1, and at 210, he returned to Variant 0. The trend in changes of beads in the "2" boxes runs: === Correlation === Depending on the strategy employed by the human player, MENACE produces a different trend on scatter graphs of wins. Using a random turn from the human player results in an almost-perfect positive trend. Playing the optimal strategy returns a slightly slower increase. The reinforcement does not create a perfect standard of wins; the algorithm will draw random uncertain conclusions each time. After the j-th round, the correlation of near-perfect play runs: 1 − D D − D ( j + 2 ) ∑ i = 0 j D ( j i + 1 ) V i {\displaystyle {1-D \over D-D^{(j+2)}}\sum _{i=0}^{j}D^{(ji+1)}V_{i}} Where Vi is the outcome (+1 is win, 0 is draw and -1 is loss) and D is the decay factor (average of past values of wins and losses). Below, Mn is the multiplier for the n-th round of the game. == Legacy == Donald Michie's MENACE proved that a computer could learn from failure and success to become good at a task. It used what would become core principles within the field of machine learning before they had been properly theorised. For example, the combination of how MENACE starts with equal numbers of types of beads in each matchbox, and how these are then selected at random, creates a learning behaviour similar to weight initialisation
Kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on kernels as weights. KDE answers a fundamental data smoothing problem where inferences about the population are made based on a finite data sample. In some fields such as signal processing and econometrics it is also termed the Parzen–Rosenblatt window method, after Emanuel Parzen and Murray Rosenblatt, who are usually credited with independently creating it in its current form. One of the famous applications of kernel density estimation is in estimating the class-conditional marginal densities of data when using a naive Bayes classifier, which can improve its prediction accuracy. == Definition == Let x = ( x 1 , x 2 , x 3 , . . . ) {\displaystyle \mathbf {x} =\left(x_{1},x_{2},x_{3},...\right)} be independent and identically distributed samples drawn from some univariate distribution with an unknown density f at any given point x. We are interested in estimating the shape of this function f. Its kernel density estimator is f ^ h ( x ) = 1 n ∑ i = 1 n K h ( x − x i ) = 1 n h ∑ i = 1 n K ( x − x i h ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}K_{h}(x-x_{i})={\frac {1}{nh}}\sum _{i=1}^{n}K{\left({\frac {x-x_{i}}{h}}\right)},} where K is the kernel — a non-negative function — and h > 0 is a smoothing parameter called the bandwidth or simply width. A kernel with subscript h is called the scaled kernel and defined as Kh(x) = 1/h K(x/h). Intuitively one wants to choose h as small as the data will allow; however, there is always a trade-off between the bias of the estimator and its variance. The choice of bandwidth is discussed in more detail below. A range of kernel functions are commonly used: uniform, triangular, biweight, triweight, Epanechnikov (parabolic), normal, and others. The Epanechnikov kernel is optimal in a mean square error sense, though the loss of efficiency is small for the kernels listed previously. Due to its convenient mathematical properties, the normal kernel is often used, which means K(x) = ϕ(x), where ϕ is the standard normal density function. The kernel density estimator then becomes f ^ h ( x ) = 1 n ∑ i = 1 n 1 h 2 π exp ( − ( x − x i ) 2 2 h 2 ) , {\displaystyle {\hat {f}}_{h}(x)={\frac {1}{n}}\sum _{i=1}^{n}{\frac {1}{h{\sqrt {2\pi }}}}\exp \left({\frac {-(x-x_{i})^{2}}{2h^{2}}}\right),} where h {\displaystyle h} is the standard deviation of the sample x {\displaystyle \mathbf {x} } . The construction of a kernel density estimate finds interpretations in fields outside of density estimation. For example, in thermodynamics, this is equivalent to the amount of heat generated when heat kernels (the fundamental solution to the heat equation) are placed at each data point locations xi. Similar methods are used to construct discrete Laplace operators on point clouds for manifold learning (e.g. diffusion map). == Example == Kernel density estimates are closely related to histograms, but can be endowed with properties such as smoothness or continuity by using a suitable kernel. The diagram below based on these 6 data points illustrates this relationship: For the histogram, first, the horizontal axis is divided into sub-intervals or bins which cover the range of the data: In this case, six bins each of width 2. Whenever a data point falls inside this interval, a box of height 1/12 is placed there. If more than one data point falls inside the same bin, the boxes are stacked on top of each other. For the kernel density estimate, normal kernels with a standard deviation of 1.5 (indicated by the red dashed lines) are placed on each of the data points xi. The kernels are summed to make the kernel density estimate (solid blue curve). The smoothness of the kernel density estimate (compared to the discreteness of the histogram) illustrates how kernel density estimates converge faster to the true underlying density for continuous random variables. == Bandwidth selection == The bandwidth of the kernel is a free parameter which exhibits a strong influence on the resulting estimate. To illustrate its effect, we take a simulated random sample from the standard normal distribution (plotted at the blue spikes in the rug plot on the horizontal axis). The grey curve is the true density (a normal density with mean 0 and variance 1). In comparison, the red curve is undersmoothed since it contains too many spurious data artifacts arising from using a bandwidth h = 0.05, which is too small. The green curve is oversmoothed since using the bandwidth h = 2 obscures much of the underlying structure. The black curve with a bandwidth of h = 0.337 is considered to be optimally smoothed since its density estimate is close to the true density. An extreme situation is encountered in the limit h → 0 {\displaystyle h\to 0} (no smoothing), where the estimate is a sum of n delta functions centered at the coordinates of analyzed samples. In the other extreme limit h → ∞ {\displaystyle h\to \infty } the estimate retains the shape of the used kernel, centered on the mean of the samples (completely smooth). The most common optimality criterion used to select this parameter is the expected L2 risk function, also termed the mean integrated squared error: MISE ( h ) = E [ ∫ ( f ^ h ( x ) − f ( x ) ) 2 d x ] {\displaystyle \operatorname {MISE} (h)=\operatorname {E} \!\left[\int \!{\left({\hat {f}}\!_{h}(x)-f(x)\right)}^{2}dx\right]} Under weak assumptions on f and K, (f is the, generally unknown, real density function), MISE ( h ) = AMISE ( h ) + o ( ( n h ) − 1 + h 4 ) {\displaystyle \operatorname {MISE} (h)=\operatorname {AMISE} (h)+{\mathcal {o}}{\left((nh)^{-1}+h^{4}\right)}} where o is the little o notation, and n the sample size (as above). The AMISE is the asymptotic MISE, i. e. the two leading terms, AMISE ( h ) = R ( K ) n h + 1 4 m 2 ( K ) 2 h 4 R ( f ″ ) {\displaystyle \operatorname {AMISE} (h)={\frac {R(K)}{nh}}+{\frac {1}{4}}m_{2}(K)^{2}h^{4}R(f'')} where R ( g ) = ∫ g ( x ) 2 d x {\textstyle R(g)=\int g(x)^{2}\,dx} for a function g, m 2 ( K ) = ∫ x 2 K ( x ) d x {\textstyle m_{2}(K)=\int x^{2}K(x)\,dx} and f ″ {\displaystyle f''} is the second derivative of f {\displaystyle f} and K {\displaystyle K} is the kernel. The minimum of this AMISE is the solution to this differential equation ∂ ∂ h AMISE ( h ) = − R ( K ) n h 2 + m 2 ( K ) 2 h 3 R ( f ″ ) = 0 {\displaystyle {\frac {\partial }{\partial h}}\operatorname {AMISE} (h)=-{\frac {R(K)}{nh^{2}}}+m_{2}(K)^{2}h^{3}R(f'')=0} or h AMISE = R ( K ) 1 / 5 m 2 ( K ) 2 / 5 R ( f ″ ) 1 / 5 n − 1 / 5 = C n − 1 / 5 {\displaystyle h_{\operatorname {AMISE} }={\frac {R(K)^{1/5}}{m_{2}(K)^{2/5}R(f'')^{1/5}}}n^{-1/5}=Cn^{-1/5}} Neither the AMISE nor the hAMISE formulas can be used directly since they involve the unknown density function f {\displaystyle f} or its second derivative f ″ {\displaystyle f''} . To overcome that difficulty, a variety of automatic, data-based methods have been developed to select the bandwidth. Several review studies have been undertaken to compare their efficacies, with the general consensus that the plug-in selectors and cross validation selectors are the most useful over a wide range of data sets. Substituting any bandwidth h which has the same asymptotic order n−1/5 as hAMISE into the AMISE gives that AMISE(h) = O(n−4/5), where O is the big O notation. It can be shown that, under weak assumptions, there cannot exist a non-parametric estimator that converges at a faster rate than the kernel estimator. Note that the n−4/5 rate is slower than the typical n−1 convergence rate of parametric methods. If the bandwidth is not held fixed, but is varied depending upon the location of either the estimate (balloon estimator) or the samples (pointwise estimator), this produces a particularly powerful method termed adaptive or variable bandwidth kernel density estimation. Bandwidth selection for kernel density estimation of heavy-tailed distributions is relatively difficult. === A rule-of-thumb bandwidth estimator === If Gaussian basis functions are used to approximate univariate data, and the underlying density being estimated is Gaussian, the optimal choice for h (that is, the bandwidth that minimises the mean integrated squared error) is: h = ( 4 σ ^ 5 3 n ) 1 / 5 ≈ 1.06 σ ^ n − 1 / 5 , {\displaystyle h={\left({\frac {4{\hat {\sigma }}^{5}}{3n}}\right)}^{1/5}\approx 1.06\,{\hat {\sigma }}\,n^{-1/5},} An h {\displaystyle h} value is considered more robust when it improves the fit for long-tailed and skewed distributions or for bimodal mixture distributions. This is often done empirically by replacing the standard deviation σ ^ {\displaystyle {\hat {\sigma }}} by the parameter A {\displaystyle A} below: A = min ( σ ^ , I Q R 1.34 ) {\displaystyle A=\min \left({\hat {\sigma }},{\frac {\mathrm {IQR} }{1.34}}\right)} where IQR is the
Video Super Resolution
RTX Video Super Resolution (RTX VSR) is a video scaling feature by Nvidia. It was released on February 28, 2023. == History == The feature was first unveiled during CES 2023 as RTX Video Super Resolution. It uses the on-board Tensor Cores to upscale browser video content in real time. Video Super Resolution was initially only available on RTX 30 and 40 series GPUs, while support for 20 series GPUs was added afterwards; it is now available on all Nvidia RTX-branded GPUs. The feature supports input resolutions from 360p to 1440p and a max output of 4K and comes without support for HDR content although that could be likely added in the future. Nvidia released RTX Video Super Resolution 1.5 with improved video quality and RTX 20 series support on October 17, 2023. == Reception == According to ComputerBase, although "the algorithm is not yet working flawlessly", the feature is "overall recommendable".
Point distribution model
The point distribution model is a model for representing the mean geometry of a shape and some statistical modes of geometric variation inferred from a training set of shapes. == Background == The point distribution model concept has been developed by Cootes, Taylor et al. and became a standard in computer vision for the statistical study of shape and for segmentation of medical images where shape priors really help interpretation of noisy and low-contrasted pixels/voxels. The latter point leads to active shape models (ASM) and active appearance models (AAM). Point distribution models rely on landmark points. A landmark is an annotating point posed by an anatomist onto a given locus for every shape instance across the training set population. For instance, the same landmark will designate the tip of the index finger in a training set of 2D hands outlines. Principal component analysis (PCA), for instance, is a relevant tool for studying correlations of movement between groups of landmarks among the training set population. Typically, it might detect that all the landmarks located along the same finger move exactly together across the training set examples showing different finger spacing for a flat-posed hands collection. == Details == First, a set of training images are manually landmarked with enough corresponding landmarks to sufficiently approximate the geometry of the original shapes. These landmarks are aligned using the generalized procrustes analysis, which minimizes the least squared error between the points. k {\displaystyle k} aligned landmarks in two dimensions are given as X = ( x 1 , y 1 , … , x k , y k ) {\displaystyle \mathbf {X} =(x_{1},y_{1},\ldots ,x_{k},y_{k})} . It's important to note that each landmark i ∈ { 1 , … k } {\displaystyle i\in \lbrace 1,\ldots k\rbrace } should represent the same anatomical location. For example, landmark #3, ( x 3 , y 3 ) {\displaystyle (x_{3},y_{3})} might represent the tip of the ring finger across all training images. Now the shape outlines are reduced to sequences of k {\displaystyle k} landmarks, so that a given training shape is defined as the vector X ∈ R 2 k {\displaystyle \mathbf {X} \in \mathbb {R} ^{2k}} . Assuming the scattering is gaussian in this space, PCA is used to compute normalized eigenvectors and eigenvalues of the covariance matrix across all training shapes. The matrix of the top d {\displaystyle d} eigenvectors is given as P ∈ R 2 k × d {\displaystyle \mathbf {P} \in \mathbb {R} ^{2k\times d}} , and each eigenvector describes a principal mode of variation along the set. Finally, a linear combination of the eigenvectors is used to define a new shape X ′ {\displaystyle \mathbf {X} '} , mathematically defined as: X ′ = X ¯ + P b {\displaystyle \mathbf {X} '={\overline {\mathbf {X} }}+\mathbf {P} \mathbf {b} } where X ¯ {\displaystyle {\overline {\mathbf {X} }}} is defined as the mean shape across all training images, and b {\displaystyle \mathbf {b} } is a vector of scaling values for each principal component. Therefore, by modifying the variable b {\displaystyle \mathbf {b} } an infinite number of shapes can be defined. To ensure that the new shapes are all within the variation seen in the training set, it is common to only allow each element of b {\displaystyle \mathbf {b} } to be within ± {\displaystyle \pm } 3 standard deviations, where the standard deviation of a given principal component is defined as the square root of its corresponding eigenvalue. PDM's can be extended to any arbitrary number of dimensions, but are typically used in 2D image and 3D volume applications (where each landmark point is R 2 {\displaystyle \mathbb {R} ^{2}} or R 3 {\displaystyle \mathbb {R} ^{3}} ). == Discussion == An eigenvector, interpreted in euclidean space, can be seen as a sequence of k {\displaystyle k} euclidean vectors associated to corresponding landmark and designating a compound move for the whole shape. Global nonlinear variation is usually well handled provided nonlinear variation is kept to a reasonable level. Typically, a twisting nematode worm is used as an example in the teaching of kernel PCA-based methods. Due to the PCA properties: eigenvectors are mutually orthogonal, form a basis of the training set cloud in the shape space, and cross at the 0 in this space, which represents the mean shape. Also, PCA is a traditional way of fitting a closed ellipsoid to a Gaussian cloud of points (whatever their dimension): this suggests the concept of bounded variation. The idea behind PDMs is that eigenvectors can be linearly combined to create an infinity of new shape instances that will 'look like' the one in the training set. The coefficients are bounded alike the values of the corresponding eigenvalues, so as to ensure the generated 2n/3n-dimensional dot will remain into the hyper-ellipsoidal allowed domain—allowable shape domain (ASD).
Matrix regularization
In the field of statistical learning theory, matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example sparsity or smoothness, that can produce stable predictive functions. For example, in the more common vector framework, Tikhonov regularization optimizes over min x ‖ A x − y ‖ 2 + λ ‖ x ‖ 2 {\displaystyle \min _{x}\left\|Ax-y\right\|^{2}+\lambda \left\|x\right\|^{2}} to find a vector x {\displaystyle x} that is a stable solution to the regression problem. When the system is described by a matrix rather than a vector, this problem can be written as min X ‖ A X − Y ‖ 2 + λ ‖ X ‖ 2 , {\displaystyle \min _{X}\left\|AX-Y\right\|^{2}+\lambda \left\|X\right\|^{2},} where the vector norm enforcing a regularization penalty on x {\displaystyle x} has been extended to a matrix norm on X {\displaystyle X} . Matrix regularization has applications in matrix completion, multivariate regression, and multi-task learning. Ideas of feature and group selection can also be extended to matrices, and these can be generalized to the nonparametric case of multiple kernel learning. == Basic definition == Consider a matrix W {\displaystyle W} to be learned from a set of examples, S = ( X i t , y i t ) {\displaystyle S=(X_{i}^{t},y_{i}^{t})} , where i {\displaystyle i} goes from 1 {\displaystyle 1} to n {\displaystyle n} , and t {\displaystyle t} goes from 1 {\displaystyle 1} to T {\displaystyle T} . Let each input matrix X i {\displaystyle X_{i}} be ∈ R D T {\displaystyle \in \mathbb {R} ^{DT}} , and let W {\displaystyle W} be of size D × T {\displaystyle D\times T} . A general model for the output y {\displaystyle y} can be posed as y i t = ⟨ W , X i t ⟩ F , {\displaystyle y_{i}^{t}=\left\langle W,X_{i}^{t}\right\rangle _{F},} where the inner product is the Frobenius inner product. For different applications the matrices X i {\displaystyle X_{i}} will have different forms, but for each of these the optimization problem to infer W {\displaystyle W} can be written as min W ∈ H E ( W ) + R ( W ) , {\displaystyle \min _{W\in {\mathcal {H}}}E(W)+R(W),} where E {\displaystyle E} defines the empirical error for a given W {\displaystyle W} , and R ( W ) {\displaystyle R(W)} is a matrix regularization penalty. The function R ( W ) {\displaystyle R(W)} is typically chosen to be convex and is often selected to enforce sparsity (using ℓ 1 {\displaystyle \ell ^{1}} -norms) and/or smoothness (using ℓ 2 {\displaystyle \ell ^{2}} -norms). Finally, W {\displaystyle W} is in the space of matrices H {\displaystyle {\mathcal {H}}} with Frobenius inner product ⟨ … ⟩ F {\displaystyle \langle \dots \rangle _{F}} . == General applications == === Matrix completion === In the problem of matrix completion, the matrix X i t {\displaystyle X_{i}^{t}} takes the form X i t = e t ⊗ e i ′ , {\displaystyle X_{i}^{t}=e_{t}\otimes e_{i}',} where ( e t ) t {\displaystyle (e_{t})_{t}} and ( e i ′ ) i {\displaystyle (e_{i}')_{i}} are the canonical basis in R T {\displaystyle \mathbb {R} ^{T}} and R D {\displaystyle \mathbb {R} ^{D}} . In this case the role of the Frobenius inner product is to select individual elements w i t {\displaystyle w_{i}^{t}} from the matrix W {\displaystyle W} . Thus, the output y {\displaystyle y} is a sampling of entries from the matrix W {\displaystyle W} . The problem of reconstructing W {\displaystyle W} from a small set of sampled entries is possible only under certain restrictions on the matrix, and these restrictions can be enforced by a regularization function. For example, it might be assumed that W {\displaystyle W} is low-rank, in which case the regularization penalty can take the form of a nuclear norm. R ( W ) = λ ‖ W ‖ ∗ = λ ∑ i | σ i | , {\displaystyle R(W)=\lambda \left\|W\right\|_{}=\lambda \sum _{i}\left|\sigma _{i}\right|,} where σ i {\displaystyle \sigma _{i}} , with i {\displaystyle i} from 1 {\displaystyle 1} to min D , T {\displaystyle \min D,T} , are the singular values of W {\displaystyle W} . === Multivariate regression === Models used in multivariate regression are parameterized by a matrix of coefficients. In the Frobenius inner product above, each matrix X {\displaystyle X} is X i t = e t ⊗ x i {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}} such that the output of the inner product is the dot product of one row of the input with one column of the coefficient matrix. The familiar form of such models is Y = X W + b {\displaystyle Y=XW+b} Many of the vector norms used in single variable regression can be extended to the multivariate case. One example is the squared Frobenius norm, which can be viewed as an ℓ 2 {\displaystyle \ell ^{2}} -norm acting either entrywise, or on the singular values of the matrix: R ( W ) = λ ‖ W ‖ F 2 = λ ∑ i ∑ j | w i j | 2 = λ Tr ( W ∗ W ) = λ ∑ i σ i 2 . {\displaystyle R(W)=\lambda \left\|W\right\|_{F}^{2}=\lambda \sum _{i}\sum _{j}\left|w_{ij}\right|^{2}=\lambda \operatorname {Tr} \left(W^{}W\right)=\lambda \sum _{i}\sigma _{i}^{2}.} In the multivariate case the effect of regularizing with the Frobenius norm is the same as the vector case; very complex models will have larger norms, and, thus, will be penalized more. === Multi-task learning === The setup for multi-task learning is almost the same as the setup for multivariate regression. The primary difference is that the input variables are also indexed by task (columns of Y {\displaystyle Y} ). The representation with the Frobenius inner product is then X i t = e t ⊗ x i t . {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}^{t}.} The role of matrix regularization in this setting can be the same as in multivariate regression, but matrix norms can also be used to couple learning problems across tasks. In particular, note that for the optimization problem min W ‖ X W − Y ‖ 2 2 + λ ‖ W ‖ 2 2 {\displaystyle \min _{W}\left\|XW-Y\right\|_{2}^{2}+\lambda \left\|W\right\|_{2}^{2}} the solutions corresponding to each column of Y {\displaystyle Y} are decoupled. That is, the same solution can be found by solving the joint problem, or by solving an isolated regression problem for each column. The problems can be coupled by adding an additional regularization penalty on the covariance of solutions min W , Ω ‖ X W − Y ‖ 2 2 + λ 1 ‖ W ‖ 2 2 + λ 2 Tr ( W T Ω − 1 W ) {\displaystyle \min _{W,\Omega }\left\|XW-Y\right\|_{2}^{2}+\lambda _{1}\left\|W\right\|_{2}^{2}+\lambda _{2}\operatorname {Tr} \left(W^{T}\Omega ^{-1}W\right)} where Ω {\displaystyle \Omega } models the relationship between tasks. This scheme can be used to both enforce similarity of solutions across tasks, and to learn the specific structure of task similarity by alternating between optimizations of W {\displaystyle W} and Ω {\displaystyle \Omega } . When the relationship between tasks is known to lie on a graph, the Laplacian matrix of the graph can be used to couple the learning problems. == Spectral regularization == Regularization by spectral filtering has been used to find stable solutions to problems such as those discussed above by addressing ill-posed matrix inversions (see for example Filter function for Tikhonov regularization). In many cases the regularization function acts on the input (or kernel) to ensure a bounded inverse by eliminating small singular values, but it can also be useful to have spectral norms that act on the matrix that is to be learned. There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix. This has been used in the context of matrix completion when the matrix in question is believed to have a restricted rank. In this case the optimization problem becomes: min ‖ W ‖ ∗ subject to W i , j = Y i j . {\displaystyle \min \left\|W\right\|_{}~~{\text{ subject to }}~~W_{i,j}=Y_{ij}.} Spectral Regularization is also used to enforce a reduced rank coefficient matrix in multivariate regression. In this setting, a reduced rank coefficient matrix can be found by keeping just the top n {\displaystyle n} singular values, but this can be extended to keep any reduced set of singular values and vectors. == Structured sparsity == Sparse optimization has become the focus of much research interest as a way to find solutions that depend on a small number of variables (see e.g. the Lasso method). In principle, entry-wise sparsity can be enforced by penalizing the entry-wise ℓ 0 {\displaystyle \ell ^{0}} -norm of the matrix, but the ℓ 0 {\displaystyle \ell ^{0}} -norm is not convex. In practice this can be implemented by convex relaxation to the ℓ 1 {\displaystyle \ell ^{1}} -norm. While entry-wise regularization with an ℓ 1 {\displaystyle \ell ^{1}} -norm will find solutions with a small number of nonzero elements, applying an ℓ 1 {