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NCAA transfer portal

The NCAA transfer portal is a National Collegiate Athletic Association (NCAA) application, database, and compliance tool that facilitates student athletes' transfers between member institutions. It is intended to bring greater transparency to the transfer process and to enable student athletes to publicize their desire to transfer. The transfer portal is an NCAA-wide database covering all three NCAA divisions, although most media coverage of the transfer portal involves its use in the top-level Division I (D-I). The portal launched on October 15, 2018. Regulations adopted in 2021 allowed student-athletes in D-I football, men's and women's basketball, men's ice hockey, and baseball to transfer schools using the portal once without sitting out a year. In 2024, the NCAA authorized athletes unlimited transfers. == Process == For Divisions I and II, once an athlete desiring to transfer informs their school; the school must enter the athlete's name in the database within two business days. Then coaches and staff from other universities may contact the athlete about potentially transferring. Before the January 2026 NCAA convention, Division III schools were allowed, but not required, to enter such a student into the portal. A proposal to require use of the portal in that division was approved at the convention. The timeline for D-III members to enter athletes into the portal differs from that of the other divisions. Athletes wishing to enter the portal must first complete an educational module. Once completed, the school has seven calendar days to enter the athlete's transfer request into the portal. == Transfer windows == On August 31, 2022, the D-I board adopted a series of changes to transfer rules, introducing the concept of transfer windows, similar to those used in professional soccer worldwide. Student-athletes who wish to take advantage of the one-time transfer rule must, under normal circumstances, enter the portal within a designated window for their sport. These windows are slightly different for each NCAA sport, but are broadly grouped by the NCAA's three athletic "seasons". At that time, the windows were as follows: Fall sports – A 45-day winter window opening the day after championship selections are made in that sport, and a spring window from May 1–15. According to the NCAA, "reasonable accommodations" would be made for participants in football's FBS and FCS championship games (respectively the College Football Playoff National Championship and Division I Football Championship Game), both of which take place in early January. Participants in those games had a 14-day window opening on the day after the championship game, as well as the spring window. Winter sports – A 60-day window opening the day after championship selections are made in that sport. Spring sports – A winter window from December 1–15, and a 45-day spring window opening the day after championship selections are made in that sport. For sports included in the NCAA Emerging Sports for Women program, transfer windows are the same as those for fully recognized NCAA sports. As with fully recognized NCAA sports, transfer windows linked to championship events open on the day after selections are made for the generally recognized championship events in emerging sports. Student-athletes whose athletic aid is reduced, canceled, or not renewed by their school, as well as those affected by a university's elimination of a sports team, may enter the transfer portal at any time without penalty. A slightly different exception applies to those undergoing a head coaching change; student-athletes so affected in sports other than Division I football can enter the portal within 30 days of the change, starting on the day after the coach's departure is announced. The coaching change window also applied to Division I football before October 2025. Less than a month after transfer windows were adopted, the Division I Council adopted a change that affected only graduate transfers. Student-athletes who are set to graduate with remaining athletic eligibility, and plan to continue competition as postgraduate students, were exempt from transfer windows. They could enter the portal at any time during the academic year, and were not subject to the standard deadlines of May 1 for fall and winter sports and July 1 for spring sports. In April 2024, graduate transfers became subject to the same deadlines as all other transfer students. This change did not affect windows for student-athletes affected by a head coaching change, a loss of athletic aid, or the discontinuation of a team. Because the Ivy League allows neither redshirting nor athletic participation by graduate students, athletes at its member schools who are set to complete four years of attendance but still have remaining athletic eligibility may enter the portal at any time during their fourth academic year of attendance. In October 2024, the Division I Council reduced transfer windows in football and basketball to a total of 30 days. For FBS and FCS football, the fall window opened for 20 days, starting on the Monday after FBS conference championship games. Participants in postseason play had a 5-day window that opened on the day after each team's final game. A 10-day spring window opened in mid-April. In men's and women's basketball, a single 30-day window opens on the day after the second round of each Division I tournament concludes. The existing exceptions regarding head coaching changes, a loss of athletic aid, or the discontinuation of a team remained in place. Almost exactly a year later, Division I adopted more significant changes to the football transfer portal for both FBS and FCS. The previous two windows were abolished and replaced by a single window that opens from January 2–16. Participants in the College Football Playoff National Championship—the only game in FBS or FCS played after the closure of the new window—receive a 5-day window that opens on the day after that game. The window for players undergoing a head coaching change was also reduced. A new window of 15 days opens five calendar days after the hiring or public announcement of a new head coach. Should a school fail to hire or publicly announce a new head coach within 30 days after the previous coach's departure, the window will open on the 31st day after departure, provided that the 31st day is no earlier than January 3. This particular window, also open for 15 days, may open at any time before June 30. No change was announced to the exceptions for those affected by a loss of athletic aid or the discontinuation of a team. == Impact on high school recruiting == Effective July 1, 2025, the NCAA Division I Board of Directors implemented new DI roster limits following the court-approved House settlement. Additionally, according to the NCAA, "NCAA rules for Division I programs will no longer include sport-specific scholarship limits." As a result, many top Division I programs, especially those in power conferences, are relying heavily on the transfer portal to bring in conference- and national-level student-athletes. This shift in recruiting focus has already been exemplified across Division I men's and women's track and field especially, beginning in the recruitment cycle for 2025 college entries. Track and field coaches formerly managing rosters of 120-plus (60-plus men and 60-plus women) are now limited to 45 per side for a total of 90 roster spots across men's and women's track and field, meaning they are recruiting fewer student-athletes out of high school and more immediately impactful scholarship-worthy student-athletes via the transfer portal. Roster limits for track and field teams are even more stringent in the Southeastern Conference (SEC): 35 men and 35 women. For high school track and field athletes seeking opportunities with top DI programs, they no longer need to display potential to be point-scorers, but demonstrate the ability to contribute immediately, often by competing at a level aligned with conference scoring standards.
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Sum of absolute transformed differences

The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison. The transform itself is often of a small block rather than the entire macroblock. For example, in x264, a series of 4×4 blocks are transformed rather than doing the more processor-intensive 16×16 transform. == Comparison to other metrics == SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instructions on most modern CPUs. The benefit of SATD is that it more accurately models the number of bits required to transmit the residual error signal. As such, it is often used in video compressors, either as a way to drive and estimate rate explicitly, such as in the Theora encoder (since 1.1 alpha2), as an optional metric used in wide motion searches, such as in the Microsoft VC-1 encoder, or as a metric used in sub-pixel refinement, such as in x264.
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Yooreeka

Yooreeka is a library for data mining, machine learning, soft computing, and mathematical analysis. The project started with the code of the book "Algorithms of the Intelligent Web". Although the term "Web" prevailed in the title, in essence, the algorithms are valuable in any software application. It covers all major algorithms and provides many examples. Yooreeka 2.x is licensed under the Apache License rather than the somewhat more restrictive LGPL (which was the license of v1.x). The library is written 100% in the Java language. == Algorithms == The following algorithms are covered: Clustering Hierarchical—Agglomerative (e.g. MST single link; ROCK) and Divisive Partitional (e.g. k-means) Classification Bayesian Decision trees Neural Networks Rule based (via Drools) Recommendations Collaborative filtering Content based Search PageRank DocRank Personalization
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Multidimensional analysis

In statistics, econometrics and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements. For example, a data set consisting of the number of wins for a single football team at each of several years is a single-dimensional (in this case, longitudinal) data set. A data set consisting of the number of wins for several football teams in a single year is also a single-dimensional (in this case, cross-sectional) data set. A data set consisting of the number of wins for several football teams over several years is a two-dimensional data set. == Higher dimensions == In many disciplines, two-dimensional data sets are also called panel data. While, strictly speaking, two- and higher-dimensional data sets are "multi-dimensional", the term "multidimensional" tends to be applied only to data sets with three or more dimensions. For example, some forecast data sets provide forecasts for multiple target periods, conducted by multiple forecasters, and made at multiple horizons. The three dimensions provide more information than can be gleaned from two-dimensional panel data sets. == Software == Computer software for MDA include Online analytical processing (OLAP) for data in relational databases, pivot tables for data in spreadsheets, and Array DBMSs for general multi-dimensional data (such as raster data) in science, engineering, and business.
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Data exploration

Data exploration is an approach similar to initial data analysis, whereby a data analyst uses visual exploration to understand what is in a dataset and the characteristics of the data, rather than through traditional data management systems. These characteristics can include size or amount of data, completeness of the data, correctness of the data, possible relationships amongst data elements or files/tables in the data. Data exploration is typically conducted using a combination of automated and manual activities. Automated activities can include data profiling or data visualization or tabular reports to give the analyst an initial view into the data and an understanding of key characteristics. This is often followed by manual drill-down or filtering of the data to identify anomalies or patterns identified through the automated actions. Data exploration can also require manual scripting and queries into the data (e.g. using languages such as SQL or R) or using spreadsheets or similar tools to view the raw data. All of these activities are aimed at creating a mental model and understanding of the data in the mind of the analyst, and defining basic metadata (statistics, structure, relationships) for the data set that can be used in further analysis. Once this initial understanding of the data is had, the data can be pruned or refined by removing unusable parts of the data (data cleansing), correcting poorly formatted elements and defining relevant relationships across datasets. This process is also known as determining data quality. Data exploration can also refer to the ad hoc querying or visualization of data to identify potential relationships or insights that may be hidden in the data and does not require to formulate assumptions beforehand. Traditionally, this had been a key area of focus for statisticians, with John Tukey being a key evangelist in the field. Today, data exploration is more widespread and is the focus of data analysts and data scientists; the latter being a relatively new role within enterprises and larger organizations. == Interactive Data Exploration == This area of data exploration has become an area of interest in the field of machine learning. This is a relatively new field and is still evolving. As its most basic level, a machine-learning algorithm can be fed a data set and can be used to identify whether a hypothesis is true based on the dataset. Common machine learning algorithms can focus on identifying specific patterns in the data. Many common patterns include regression and classification or clustering, but there are many possible patterns and algorithms that can be applied to data via machine learning. By employing machine learning, it is possible to find patterns or relationships in the data that would be difficult or impossible to find via manual inspection, trial and error or traditional exploration techniques. == Software == Trifacta – a data preparation and analysis platform Paxata – self-service data preparation software Alteryx – data blending and advanced data analytics software Microsoft Power BI - interactive visualization and data analysis tool OpenRefine - a standalone open source desktop application for data clean-up and data transformation Tableau software – interactive data visualization software
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Dynamic time warping

In time series analysis, dynamic time warping (DTW) is an algorithm for measuring similarity between two temporal sequences, which may vary in speed. For instance, similarities in walking could be detected using DTW, even if one person was walking faster than the other, or if there were accelerations and decelerations during the course of an observation. DTW has been applied to temporal sequences of video, audio, and graphics data — indeed, any data that can be turned into a one-dimensional sequence can be analyzed with DTW. A well-known application has been automatic speech recognition, to cope with different speaking speeds. Other applications include speaker recognition and online signature recognition. It can also be used in partial shape matching applications. In general, DTW is a method that calculates an optimal match between two given sequences (e.g. time series) with certain restriction and rules: Every index from the first sequence must be matched with one or more indices from the other sequence, and vice versa The first index from the first sequence must be matched with the first index from the other sequence (but it does not have to be its only match) The last index from the first sequence must be matched with the last index from the other sequence (but it does not have to be its only match) The mapping of the indices from the first sequence to indices from the other sequence must be monotonically increasing, and vice versa, i.e. if j > i {\displaystyle j>i} are indices from the first sequence, then there must not be two indices l > k {\displaystyle l>k} in the other sequence, such that index i {\displaystyle i} is matched with index l {\displaystyle l} and index j {\displaystyle j} is matched with index k {\displaystyle k} , and vice versa We can plot each match between the sequences 1 : M {\displaystyle 1:M} and 1 : N {\displaystyle 1:N} as a path in a M × N {\displaystyle M\times N} matrix from ( 1 , 1 ) {\displaystyle (1,1)} to ( M , N ) {\displaystyle (M,N)} , such that each step is one of ( 0 , 1 ) , ( 1 , 0 ) , ( 1 , 1 ) {\displaystyle (0,1),(1,0),(1,1)} . In this formulation, we see that the number of possible matches is the Delannoy number. The optimal match is denoted by the match that satisfies all the restrictions and the rules and that has the minimal cost, where the cost is computed as the sum of absolute differences, for each matched pair of indices, between their values. The sequences are "warped" non-linearly in the time dimension to determine a measure of their similarity independent of certain non-linear variations in the time dimension. This sequence alignment method is often used in time series classification. Although DTW measures a distance-like quantity between two given sequences, it doesn't guarantee the triangle inequality to hold. In addition to a similarity measure between the two sequences (a so called "warping path" is produced), by warping according to this path the two signals may be aligned in time. The signal with an original set of points X(original), Y(original) is transformed to X(warped), Y(warped). This finds applications in genetic sequence and audio synchronisation. In a related technique sequences of varying speed may be averaged using this technique see the average sequence section. This is conceptually very similar to the Needleman–Wunsch algorithm. == Implementation == This example illustrates the implementation of the dynamic time warping algorithm when the two sequences s and t are strings of discrete symbols. For two symbols x and y, d ( x , y ) {\displaystyle d(x,y)} is a distance between the symbols, e.g., d ( x , y ) = | x − y | {\displaystyle d(x,y)=|x-y|} . int DTWDistance(s: array [1..n], t: array [1..m]) { DTW := array [0..n, 0..m] for i := 0 to n for j := 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := 1 to m cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } where DTW[i, j] is the distance between s[1:i] and t[1:j] with the best alignment. We sometimes want to add a locality constraint. That is, we require that if s[i] is matched with t[j], then | i − j | {\displaystyle |i-j|} is no larger than w, a window parameter. We can easily modify the above algorithm to add a locality constraint (differences marked). However, the above given modification works only if | n − m | {\displaystyle |n-m|} is no larger than w, i.e. the end point is within the window length from diagonal. In order to make the algorithm work, the window parameter w must be adapted so that | n − m | ≤ w {\displaystyle |n-m|\leq w} (see the line marked with () in the code). int DTWDistance(s: array [1..n], t: array [1..m], w: int) { DTW := array [0..n, 0..m] w := max(w, abs(n-m)) // adapt window size () for i := 0 to n for j:= 0 to m DTW[i, j] := infinity DTW[0, 0] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) DTW[i, j] := 0 for i := 1 to n for j := max(1, i-w) to min(m, i+w) cost := d(s[i], t[j]) DTW[i, j] := cost + minimum(DTW[i-1, j ], // insertion DTW[i , j-1], // deletion DTW[i-1, j-1]) // match return DTW[n, m] } == Warping properties == The DTW algorithm produces a discrete matching between existing elements of one series to another. In other words, it does not allow time-scaling of segments within the sequence. Other methods allow continuous warping. For example, Correlation Optimized Warping (COW) divides the sequence into uniform segments that are scaled in time using linear interpolation, to produce the best matching warping. The segment scaling causes potential creation of new elements, by time-scaling segments either down or up, and thus produces a more sensitive warping than DTW's discrete matching of raw elements. == Complexity == The time complexity of the DTW algorithm is O ( N M ) {\displaystyle O(NM)} , where N {\displaystyle N} and M {\displaystyle M} are the lengths of the two input sequences. The 50 years old quadratic time bound was broken in 2016: an algorithm due to Gold and Sharir enables computing DTW in O ( N 2 / log ⁡ log ⁡ N ) {\displaystyle O({N^{2}}/\log \log N)} time and space for two input sequences of length N {\displaystyle N} . This algorithm can also be adapted to sequences of different lengths. Despite this improvement, it was shown that a strongly subquadratic running time of the form O ( N 2 − ϵ ) {\displaystyle O(N^{2-\epsilon })} for some ϵ > 0 {\displaystyle \epsilon >0} cannot exist unless the Strong exponential time hypothesis fails. While the dynamic programming algorithm for DTW requires O ( N M ) {\displaystyle O(NM)} space in a naive implementation, the space consumption can be reduced to O ( min ( N , M ) ) {\displaystyle O(\min(N,M))} using Hirschberg's algorithm. == Fast computation == Fast techniques for computing DTW include PrunedDTW, SparseDTW, FastDTW, and the MultiscaleDTW. A common task, retrieval of similar time series, can be accelerated by using lower bounds such as LB_Keogh, LB_Improved, or LB_Petitjean. However, the Early Abandon and Pruned DTW algorithm reduces the degree of acceleration that lower bounding provides and sometimes renders it ineffective. In a survey, Wang et al. reported slightly better results with the LB_Improved lower bound than the LB_Keogh bound, and found that other techniques were inefficient. Subsequent to this survey, the LB_Enhanced bound was developed that is always tighter than LB_Keogh while also being more efficient to compute. LB_Petitjean is the tightest known lower bound that can be computed in linear time. == Average sequence == Averaging for dynamic time warping is the problem of finding an average sequence for a set of sequences. NLAAF is an exact method to average two sequences using DTW. For more than two sequences, the problem is related to that of multiple alignment and requires heuristics. DBA is currently a reference method to average a set of sequences consistently with DTW. COMASA efficiently randomizes the search for the average sequence, using DBA as a local optimization process. == Supervised learning == A nearest-neighbour classifier can achieve state-of-the-art performance when using dynamic time warping as a distance measure. == Amerced Dynamic Time Warping == Amerced Dynamic Time Warping (ADTW) is a variant of DTW designed to better control DTW's permissiveness in the alignments that it allows. The windows that classical DTW uses to constrain alignments introduce a step function. Any warping of the path is allowed within the window and none beyond it. In contrast, ADTW employs an additive penalty that is incurred each time that the path is warped. Any amount of warping is allowed, but each warping action incurs a direct penalty. ADTW significantly outperforms DTW with windowing when applied as a nearest neighbor classifier on a set of benchmark time series classification tasks. == Alternative approaches == In functional data analysis, time series are regarde
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Types of artificial neural networks

Types of neural networks (NN) include a family of techniques. The simplest types have static components, including number of units, number of layers, unit weights and topology. Dynamic NNs evolve via learning. Some types allow/require learning to be "supervised" by the operator, while others operate independently. Some types operate purely in hardware, while others are purely software and run on general purpose computers. The main types are: Transformers: these use attention to analyze every token in the input stream against every other token in the stream. That technique has enabled neural networks to reach the general public via chatbots, code generators and many other forms. Convolutional neural networks (CNN): a FNN that uses kernels and regularization to evade problems in prior generations of NNs. They are typically used to analyze visual and other two-dimensional data. Generative adversarial networks set networks (of varying structure) against each other, each trying to push the other(s) to produce better results such as winning a game or to deceive the opponent about the authenticity of an input. == Feedforward == In feedforward neural networks the information moves from the input to output directly in every layer. There can be hidden layers with or without cycles/loops to sequence inputs. Feedforward networks can be constructed with various types of units, such as binary McCulloch–Pitts neurons, the simplest of which is the perceptron. Continuous neurons, frequently with sigmoidal activation, are used in the context of backpropagation. == Group method of data handling == The Group Method of Data Handling (GMDH) features fully automatic structural and parametric model optimization. The node activation functions are Kolmogorov–Gabor polynomials that permit additions and multiplications. It uses a deep multilayer perceptron with eight layers. It is a supervised learning network that grows layer by layer, where each layer is trained by regression analysis. Useless items are detected using a validation set, and pruned through regularization. The size and depth of the resulting network depends on the task. == Autoencoder == An autoencoder, autoassociator or Diabolo network is similar to the multilayer perceptron (MLP) – with an input layer, an output layer and one or more hidden layers connecting them. However, the output layer has the same number of units as the input layer. Its purpose is to reconstruct its own inputs (instead of emitting a target value). Therefore, autoencoders are unsupervised learning models. An autoencoder is used for unsupervised learning of efficient codings, typically for the purpose of dimensionality reduction and for learning generative models of data. == Probabilistic == A probabilistic neural network (PNN) is a four-layer feedforward neural network. The layers are Input, hidden pattern, hidden summation, and output. In the PNN algorithm, the parent probability distribution function (PDF) of each class is approximated by a Parzen window and a non-parametric function. Then, using PDF of each class, the class probability of a new input is estimated and Bayes’ rule is employed to allocate it to the class with the highest posterior probability. It was derived from the Bayesian network and a statistical algorithm called Kernel Fisher discriminant analysis. It is used for classification and pattern recognition. == Time delay == A time delay neural network (TDNN) is a feedforward architecture for sequential data that recognizes features independent of sequence position. In order to achieve time-shift invariance, delays are added to the input so that multiple data points (points in time) are analyzed together. It usually forms part of a larger pattern recognition system. It has been implemented using a perceptron network whose connection weights were trained with back propagation (supervised learning). == Convolutional == A convolutional neural network (CNN, or ConvNet or shift invariant or space invariant) is a class of deep network, composed of one or more convolutional layers with fully connected layers (matching those in typical ANNs) on top. It uses tied weights and pooling layers. In particular, max-pooling. It is often structured via Fukushima's convolutional architecture. They are variations of multilayer perceptrons that use minimal preprocessing. This architecture allows CNNs to take advantage of the 2D structure of input data. Its unit connectivity pattern is inspired by the organization of the visual cortex. Units respond to stimuli in a restricted region of space known as the receptive field. Receptive fields partially overlap, over-covering the entire visual field. Unit response can be approximated mathematically by a convolution operation. CNNs are suitable for processing visual and other two-dimensional data. They have shown superior results in both image and speech applications. They can be trained with standard backpropagation. CNNs are easier to train than other regular, deep, feed-forward neural networks and have many fewer parameters to estimate. Capsule Neural Networks (CapsNet) add structures called capsules to a CNN and reuse output from several capsules to form more stable (with respect to various perturbations) representations. Examples of applications in computer vision include DeepDream and robot navigation. They have wide applications in image and video recognition, recommender systems and natural language processing. == Deep stacking network == A deep stacking network (DSN) (deep convex network) is based on a hierarchy of blocks of simplified neural network modules. It was introduced in 2011 by Deng and Yu. It formulates the learning as a convex optimization problem with a closed-form solution, emphasizing the mechanism's similarity to stacked generalization. Each DSN block is a simple module that is easy to train by itself in a supervised fashion without backpropagation for the entire blocks. Each block consists of a simplified multi-layer perceptron (MLP) with a single hidden layer. The hidden layer h has logistic sigmoidal units, and the output layer has linear units. Connections between these layers are represented by weight matrix U; input-to-hidden-layer connections have weight matrix W. Target vectors t form the columns of matrix T, and the input data vectors x form the columns of matrix X. The matrix of hidden units is H = σ ( W T X ) {\displaystyle {\boldsymbol {H}}=\sigma ({\boldsymbol {W}}^{T}{\boldsymbol {X}})} . Modules are trained in order, so lower-layer weights W are known at each stage. The function performs the element-wise logistic sigmoid operation. Each block estimates the same final label class y, and its estimate is concatenated with original input X to form the expanded input for the next block. Thus, the input to the first block contains the original data only, while downstream blocks' input adds the output of preceding blocks. Then learning the upper-layer weight matrix U given other weights in the network can be formulated as a convex optimization problem: min U T f = ‖ U T H − T ‖ F 2 , {\displaystyle \min _{U^{T}}f=\|{\boldsymbol {U}}^{T}{\boldsymbol {H}}-{\boldsymbol {T}}\|_{F}^{2},} which has a closed-form solution. Unlike other deep architectures, such as DBNs, the goal is not to discover the transformed feature representation. The structure of the hierarchy of this kind of architecture makes parallel learning straightforward, as a batch-mode optimization problem. In purely discriminative tasks, DSNs outperform conventional DBNs. === Tensor deep stacking networks === This architecture is a DSN extension. It offers two important improvements: it uses higher-order information from covariance statistics, and it transforms the non-convex problem of a lower-layer to a convex sub-problem of an upper-layer. TDSNs use covariance statistics in a bilinear mapping from each of two distinct sets of hidden units in the same layer to predictions, via a third-order tensor. While parallelization and scalability are not considered seriously in conventional DNNs, all learning for DSNs and TDSNs is done in batch mode, to allow parallelization. Parallelization allows scaling the design to larger (deeper) architectures and data sets. The basic architecture is suitable for diverse tasks such as classification and regression. == Physics-informed == Such a neural network is designed for the numerical solution of mathematical equations, such as differential, integral, delay, fractional and others. As input parameters, PINN accepts variables (spatial, temporal, and others), transmits them through the network block. At the output, it produces an approximate solution and substitutes it into the mathematical model, considering the initial and boundary conditions. If the solution does not satisfy the required accuracy, one uses the backpropagation and rectify the solution. Besides PINN, other architectures have been developed to produce surrogate models for scientific comput
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Autoencoder

An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg ⁡ min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-
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International Clinical Trials Registry Platform

The International Clinical Trials Registry Platform (ICTRP) is a platform for the registration of clinical trials operated by the World Health Organization. The ICTRP combines data from multiple cooperating clinical trials registries to generate a global view of clinical trials worldwide, with a search portal that allows access to the entire dataset. It requires a minimum standard set of database fields, the WHO Trial Registration Data Set, to be present for a trial to be registered. All entries are given a Universal Trial Number (UTN) that identifies them uniquely. The organization has sought to assist various national governments in establishing their own clinical trials databases. It combines data from the following primary registries and data providers: Australian New Zealand Clinical Trials Registry (ANZCTR) Brazilian Clinical Trials Registry (ReBec) Chinese Clinical Trial Registry (ChiCTR) Clinical Research Information Service (CRiS), Republic of Korea ClinicalTrials.gov Clinical Trials Information System (CTIS), European Medicines Agency Clinical Trials Registry - India (CTRI) Cuban Public Registry of Clinical Trials (RPCEC) EU Clinical Trials Register (EU-CTR) German Clinical Trials Register (DRKS) Iranian Registry of Clinical Trials (IRCT) ISRCTN (UK) International Traditional Medicine Clinical Trial Registry (ITMCTR) Japan Registry of Clinical Trials (jRCT) Japan Primary Registries Network (JPRN) Lebanese Clinical Trials Registry (LBCTR) Overview of Medical Research in the Netherlands (OMON) Thai Clinical Trials Registry (TCTR) Pan African Clinical Trial Registry (PACTR) Peruvian Clinical Trial Registry (REPEC) Sri Lanka Clinical Trials Registry (SLCTR)
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Receptron

The receptron (short for "reservoir perceptron") is a neuromorphic data processing model — specifically neuromorphic computing — that generalizes the traditional perceptron, by incorporating non-linear interactions between inputs. Unlike classical perceptron, which rely on linearly independent weights, the receptron leverages complexity in physical substrates, such as the electric conduction properties of nanostructured materials or optical speckle fields, to perform classification tasks. The receptron bridges unconventional computing and neural network principles, enabling solutions that do not require the training approaches typical of artificial neural networks based on the perceptron model. == Algorithm == The receptron is an algorithm for supervised learning of binary classifiers, so a classification algorithm that makes its predictions based on a predictor function, combining a set of weights with the feature vector. The mathematical model is based on the sum of inputs with non-linear interactions: S = ∑ k = 1 n x j w ~ j ( x → ) | S ∈ R {\displaystyle S=\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})|S\in R} (1) where j ∈ [ 1 , n ] {\displaystyle j\in [1,n]} and w ~ j {\displaystyle {\widetilde {w}}_{j}} are non-linear weight functions depending on the inputs, x → {\displaystyle {\vec {x}}} . Nonlinearity will typically make the system extremely complex, and allowing for the solution of problems not solvable through the simpler rules of a linear system, such as the perceptron or McCulloch Pitts neurons, which is based on the sum of linearly independent weights: S = ∑ k = 1 n x j w j p {\displaystyle S=\sum _{k=1}^{n}x_{j}w_{j}^{p}} (2) where w j {\displaystyle w_{j}} are constant real values. A consequence of this simplicity is the limitation to linearly separable functions, which necessitates multi-layer architectures and training algorithms like backpropagation As in the perceptron case, the summation in Eq. 1 origins the activation of the receptron output through the thresholding process, Y ( x 1 , . . . , x n ) = { 1 if S > th 0 if S ≤ th {\displaystyle Y(x_{1},...,x_{n})={\begin{cases}1&{\text{if }}S>{\text{th}}\\0&{\text{if }}S\leq {\text{th}}\end{cases}}} (3) where th is a constant threshold parameter. Equation 3 can be written by using the Heaviside step function. The weight functions w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} can be written with a finite number of parameters w j 1 . . . j n {\displaystyle w_{j_{1}...j_{n}}} , simplifying the model representation. One can Taylor-expand w ~ ( x → ) {\displaystyle {\widetilde {w}}({\vec {x}})} and use the idempotency of Boolean variables ( x j ) q = x j ∀ q ≥ 1 {\displaystyle (x_{j})^{q}=x_{j}\forall q\geq 1} such that S ′ = b + ∑ k = 1 n x j w ~ j ( x → ) {\displaystyle S'=b+\sum _{k=1}^{n}x_{j}{\widetilde {w}}_{j}({\vec {x}})} can be written as S ′ ( x → ) = b + ∑ j w j x j + ∑ j < k w j k x j x k + ∑ j < k < l w j k l x j x k x l + . . . {\displaystyle S'({\vec {x}})=b+\sum _{j}w_{j}x_{j}+\sum _{j Read more →
Bondy's theorem

In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972. == Statement == The theorem is as follows: Let X be a set with n elements and let A1, A2, ..., An be distinct subsets of X. Then there exists a subset S of X with n − 1 elements such that the sets Ai ∩ S are all distinct. In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct, we can remove one column such that the rows of the resulting n × (n − 1) matrix are distinct. == Example == Consider the 4 × 4 matrix [ 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 ] {\displaystyle {\begin{bmatrix}1&1&0&1\\0&1&0&1\\0&0&1&1\\0&1&1&0\end{bmatrix}}} where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix [ 1 0 1 1 0 1 0 1 1 1 1 0 ] {\displaystyle {\begin{bmatrix}1&0&1\\1&0&1\\0&1&1\\1&1&0\end{bmatrix}}} no longer has this property: the first row is identical to the second row. Nevertheless, by Bondy's theorem we know that we can always find a column that can be deleted without introducing any identical rows. In this case, we can delete the third column: all rows of the 3 × 4 matrix [ 1 1 1 0 1 1 0 0 1 0 1 0 ] {\displaystyle {\begin{bmatrix}1&1&1\\0&1&1\\0&0&1\\0&1&0\end{bmatrix}}} are distinct. Another possibility would have been deleting the fourth column. == Learning theory application == From the perspective of computational learning theory, Bondy's theorem can be rephrased as follows: Let C be a concept class over a finite domain X. Then there exists a subset S of X with the size at most |C| − 1 such that S is a witness set for every concept in C. This implies that every finite concept class C has its teaching dimension bounded by |C| − 1.
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PVLV

The primary value learned value (PVLV) model is a possible explanation for the reward-predictive firing properties of dopamine (DA) neurons. It simulates behavioral and neural data on Pavlovian conditioning and the midbrain dopaminergic neurons that fire in proportion to unexpected rewards. It is an alternative to the temporal-differences (TD) algorithm. It is used as part of Leabra.
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SAP BTP

SAP Business Technology Platform (SAP BTP) is a platform as a service developed by SAP SE that offers a suite of services including database and data management, AI, analytics, application development, automation and integration all running on one unified platform. == Overview == SAP BTP is made up of four components: Application development and automation: to create applications or extend existing applications. Data and analytics: to access and analyze data across SAP and third-party systems using multi-cloud architecture. Integration: to integrate and connect applications and data. Artificial Intelligence (AI): to access large language models (LLMs) to develop AI. == History == SAP BTP was introduced as part of the SAP strategy to unify its portfolio and cloud offerings under a single platform. The platform was evolved from earlier initiatives such as SAP Cloud Platform and now serves as the central hub for cloud, data, analytics, integration and AI technologies. Initially unveiled as "SAP NetWeaver Cloud" belonging to the SAP HANA Cloud portfolio on October 16, 2012 the cloud platform was reintroduced with the new name "SAP HANA Cloud Platform" on May 13, 2013 as the foundation for SAP cloud products, including the SAP BusinessObjects Cloud. Adoption of the SAP HANA Cloud Platform in 2015 stood at over 4000 customers and 500 partners. In 2016, SAP and Apple Inc. partnered to develop mobile applications on iOS using cloud-based software development kits (SDKs) for the SAP Cloud Platform. On February 27, 2017, SAP HANA Cloud Platform was renamed "SAP Cloud Platform" at the Mobile World Congress. On January 18, 2021, the name "SAP Cloud Platform" was retired from the SAP product portfolio to support SAP BTP. As of October 2024, SAP states that SAP BTP is used by more than 27,000 customers and more than 2,800 partners. Recently, SAP Business One has worked on improving the functionalities of BTP to cater for the demands of digital transformation. The platform offers comprehensive services in AI, application development, automation, integration, data management, and analytics.
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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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Low-rank approximation

In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization variable), subject to a constraint that the approximating matrix has reduced rank. The problem is used for mathematical modeling and data compression. The rank constraint is related to a constraint on the complexity of a model that fits the data. In applications, often there are other constraints on the approximating matrix apart from the rank constraint, e.g., non-negativity and Hankel structure. Low-rank approximation is closely related to numerous other techniques, including principal component analysis, factor analysis, total least squares, latent semantic analysis, orthogonal regression, and dynamic mode decomposition. == Definition == Given structure specification S : R n p → R m × n {\displaystyle {\mathcal {S}}:\mathbb {R} ^{n_{p}}\to \mathbb {R} ^{m\times n}} , vector of structure parameters p ∈ R n p {\displaystyle p\in \mathbb {R} ^{n_{p}}} , norm ‖ ⋅ ‖ {\displaystyle \|\cdot \|} , and desired rank r {\displaystyle r} , minimize over p ^ ‖ p − p ^ ‖ subject to rank ⁡ ( S ( p ^ ) ) ≤ r . {\displaystyle {\text{minimize}}\quad {\text{over }}{\widehat {p}}\quad \|p-{\widehat {p}}\|\quad {\text{subject to}}\quad \operatorname {rank} {\big (}{\mathcal {S}}({\widehat {p}}){\big )}\leq r.} == Applications == Linear system identification, in which case the approximating matrix is Hankel structured. Machine learning, in which case the approximating matrix is nonlinearly structured. Recommender systems, in which cases the data matrix has missing values and the approximation is categorical. Distance matrix completion, in which case there is a positive definiteness constraint. Natural language processing, in which case the approximation is nonnegative. Computer algebra, in which case the approximation is Sylvester structured. Matrix product states, in which case the approximation is usually rescaled to have fixed Frobenius norm. == Basic low-rank approximation problem == The unstructured problem with fit measured by the Frobenius norm, i.e., minimize over D ^ ‖ D − D ^ ‖ F subject to rank ⁡ ( D ^ ) ≤ r {\displaystyle {\text{minimize}}\quad {\text{over }}{\widehat {D}}\quad \|D-{\widehat {D}}\|_{\text{F}}\quad {\text{subject to}}\quad \operatorname {rank} {\big (}{\widehat {D}}{\big )}\leq r} has an analytic solution in terms of the singular value decomposition of the data matrix. The result is referred to as the matrix approximation lemma or Eckart–Young–Mirsky theorem. This problem was originally solved by Erhard Schmidt in the infinite dimensional context of integral operators (although his methods easily generalize to arbitrary compact operators on Hilbert spaces) and later rediscovered by C. Eckart and G. Young. L. Mirsky generalized the result to arbitrary unitarily invariant norms. Let D = U Σ V ⊤ ∈ R m × n , m ≥ n {\displaystyle D=U\Sigma V^{\top }\in \mathbb {R} ^{m\times n},\quad m\geq n} be the singular value decomposition of D {\displaystyle D} , where Σ =: diag ⁡ ( σ 1 , … , σ r ) {\displaystyle \Sigma =:\operatorname {diag} (\sigma _{1},\ldots ,\sigma _{r})} , where r ≤ min { m , n } = n {\displaystyle r\leq \min\{m,n\}=n} , is the m × n {\displaystyle m\times n} rectangular diagonal matrix with r {\displaystyle r} non-zero singular values σ 1 ≥ … ≥ σ r > σ r + 1 = … = σ n = 0 {\displaystyle \sigma _{1}\geq \ldots \geq \sigma _{r}>\sigma _{r+1}=\ldots =\sigma _{n}=0} . For a given k ∈ { 1 , … , r } {\displaystyle k\in \{1,\dots ,r\}} , partition U {\displaystyle U} , Σ {\displaystyle \Sigma } , and V {\displaystyle V} as follows: U =: [ U 1 U 2 ] , Σ =: [ Σ 1 0 0 Σ 2 ] , and V =: [ V 1 V 2 ] , {\displaystyle U=:{\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}},\quad \Sigma =:{\begin{bmatrix}\Sigma _{1}&0\\0&\Sigma _{2}\end{bmatrix}},\quad {\text{and}}\quad V=:{\begin{bmatrix}V_{1}&V_{2}\end{bmatrix}},} where U 1 {\displaystyle U_{1}} is m × k {\displaystyle m\times k} , Σ 1 {\displaystyle \Sigma _{1}} is k × k {\displaystyle k\times k} , and V 1 {\displaystyle V_{1}} is n × k {\displaystyle n\times k} . Then the rank k {\displaystyle k} matrix D ^ ∗ := U 1 Σ 1 V 1 ⊤ , {\displaystyle {\widehat {D}}^{}:=U_{1}\Sigma _{1}V_{1}^{\top },} obtained from the truncated singular value decomposition is such that ‖ D − D ^ ∗ ‖ F = min rank ⁡ ( D ^ ) ≤ k ‖ D − D ^ ‖ F = σ k + 1 2 + ⋯ + σ r 2 . {\displaystyle \|D-{\widehat {D}}^{}\|_{\text{F}}=\min _{\operatorname {rank} ({\widehat {D}})\leq k}\|D-{\widehat {D}}\|_{\text{F}}={\sqrt {\sigma _{k+1}^{2}+\cdots +\sigma _{r}^{2}}}.} The minimizer D ^ ∗ {\displaystyle {\widehat {D}}^{}} is unique if and only if σ k > σ k + 1 {\displaystyle \sigma _{k}>\sigma _{k+1}} . == Proof of Eckart–Young–Mirsky theorem (for spectral norm) == Let A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} be a real (possibly rectangular) matrix with m ≤ n {\displaystyle m\leq n} . Suppose that A = U Σ V ⊤ {\displaystyle A=U\Sigma V^{\top }} is the singular value decomposition of A {\displaystyle A} . Recall that U {\displaystyle U} and V {\displaystyle V} are orthogonal matrices, and Σ {\displaystyle \Sigma } is an m × n {\displaystyle m\times n} diagonal matrix with entries ( σ 1 , σ 2 , ⋯ , σ m ) {\displaystyle (\sigma _{1},\sigma _{2},\cdots ,\sigma _{m})} such that σ 1 ≥ σ 2 ≥ ⋯ ≥ σ m ≥ 0 {\displaystyle \sigma _{1}\geq \sigma _{2}\geq \cdots \geq \sigma _{m}\geq 0} . We claim that the best rank- k {\displaystyle k} approximation to A {\displaystyle A} in the spectral norm, denoted by ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} , is given by A k := ∑ i = 1 k σ i u i v i ⊤ {\displaystyle A_{k}:=\sum _{i=1}^{k}\sigma _{i}u_{i}v_{i}^{\top }} where u i {\displaystyle u_{i}} and v i {\displaystyle v_{i}} denote the i {\displaystyle i} th column of U {\displaystyle U} and V {\displaystyle V} , respectively. First, note that we have ‖ A − A k ‖ 2 = ‖ ∑ i = 1 n σ i u i v i ⊤ − ∑ i = 1 k σ i u i v i ⊤ ‖ 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ 2 = σ k + 1 {\displaystyle \|A-A_{k}\|_{2}=\left\|\sum _{i=1}^{\color {red}{n}}\sigma _{i}u_{i}v_{i}^{\top }-\sum _{i=1}^{\color {red}{k}}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{2}=\left\|\sum _{i=\color {red}{k+1}}^{n}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{2}=\sigma _{k+1}} Therefore, we need to show that if B k = X Y ⊤ {\displaystyle B_{k}=XY^{\top }} where X {\displaystyle X} and Y {\displaystyle Y} have k {\displaystyle k} columns then ‖ A − A k ‖ 2 = σ k + 1 ≤ ‖ A − B k ‖ 2 {\displaystyle \|A-A_{k}\|_{2}=\sigma _{k+1}\leq \|A-B_{k}\|_{2}} . Since Y {\displaystyle Y} has k {\displaystyle k} columns, then there must be a nontrivial linear combination of the first k + 1 {\displaystyle k+1} columns of V {\displaystyle V} , i.e., w = γ 1 v 1 + ⋯ + γ k + 1 v k + 1 , {\displaystyle w=\gamma _{1}v_{1}+\cdots +\gamma _{k+1}v_{k+1},} such that Y ⊤ w = 0 {\displaystyle Y^{\top }w=0} . Without loss of generality, we can scale w {\displaystyle w} so that ‖ w ‖ 2 = 1 {\displaystyle \|w\|_{2}=1} or (equivalently) γ 1 2 + ⋯ + γ k + 1 2 = 1 {\displaystyle \gamma _{1}^{2}+\cdots +\gamma _{k+1}^{2}=1} . Therefore, ‖ A − B k ‖ 2 2 ≥ ‖ ( A − B k ) w ‖ 2 2 = ‖ A w ‖ 2 2 = γ 1 2 σ 1 2 + ⋯ + γ k + 1 2 σ k + 1 2 ≥ σ k + 1 2 . {\displaystyle \|A-B_{k}\|_{2}^{2}\geq \|(A-B_{k})w\|_{2}^{2}=\|Aw\|_{2}^{2}=\gamma _{1}^{2}\sigma _{1}^{2}+\cdots +\gamma _{k+1}^{2}\sigma _{k+1}^{2}\geq \sigma _{k+1}^{2}.} The result follows by taking the square root of both sides of the above inequality. == Proof of Eckart–Young–Mirsky theorem (for Frobenius norm) == Let A ∈ R m × n {\displaystyle A\in \mathbb {R} ^{m\times n}} be a real (possibly rectangular) matrix with m ≤ n {\displaystyle m\leq n} . Suppose that A = U Σ V ⊤ {\displaystyle A=U\Sigma V^{\top }} is the singular value decomposition of A {\displaystyle A} . We claim that the best rank k {\displaystyle k} approximation to A {\displaystyle A} in the Frobenius norm, denoted by ‖ ⋅ ‖ F {\displaystyle \|\cdot \|_{F}} , is given by A k = ∑ i = 1 k σ i u i v i ⊤ {\displaystyle A_{k}=\sum _{i=1}^{k}\sigma _{i}u_{i}v_{i}^{\top }} where u i {\displaystyle u_{i}} and v i {\displaystyle v_{i}} denote the i {\displaystyle i} th column of U {\displaystyle U} and V {\displaystyle V} , respectively. First, note that we have ‖ A − A k ‖ F 2 = ‖ ∑ i = k + 1 n σ i u i v i ⊤ ‖ F 2 = ∑ i = k + 1 n σ i 2 {\displaystyle \|A-A_{k}\|_{F}^{2}=\left\|\sum _{i=k+1}^{n}\sigma _{i}u_{i}v_{i}^{\top }\right\|_{F}^{2}=\sum _{i=k+1}^{n}\sigma _{i}^{2}} Therefore, we need to show that if B k = X Y ⊤ {\displaystyle B_{k}=XY^{\top }} where X {\displaystyle X} and Y {\displaystyle Y} have k {\displaystyle k} columns then ‖ A − A k ‖ F 2 = ∑ i = k + 1 n σ i 2 ≤ ‖ A − B k ‖ F 2 . {\displaystyle \|A-A_{k}\|_{F}^{2}=\sum _{i=k+1}^{n}\sigma _{i}^{2}\leq \|A-B_{k}\|_{F}^{2}.} By the triangle inequality with the spectral norm
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