Birkhoff algorithm

Birkhoff's algorithm (also called Birkhoff-von-Neumann algorithm) is an algorithm for decomposing a bistochastic matrix into a convex combination of permutation matrices. It was published by Garrett Birkhoff in 1946. It has many applications. One such application is for the problem of fair random assignment: given a randomized allocation of items, Birkhoff's algorithm can decompose it into a lottery on deterministic allocations. == Terminology == A bistochastic matrix (also called: doubly-stochastic) is a matrix in which all elements are greater than or equal to 0 and the sum of the elements in each row and column equals 1. An example is the following 3-by-3 matrix: ( 0.2 0.3 0.5 0.6 0.2 0.2 0.2 0.5 0.3 ) {\displaystyle {\begin{pmatrix}0.2&0.3&0.5\\0.6&0.2&0.2\\0.2&0.5&0.3\end{pmatrix}}} A permutation matrix is a special case of a bistochastic matrix, in which each element is either 0 or 1 (so there is exactly one "1" in each row and each column). An example is the following 3-by-3 matrix: ( 0 1 0 0 0 1 1 0 0 ) {\displaystyle {\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}}} A Birkhoff decomposition (also called: Birkhoff-von-Neumann decomposition) of a bistochastic matrix is a presentation of it as a sum of permutation matrices with non-negative weights. For example, the above matrix can be presented as the following sum: 0.2 ( 0 1 0 0 0 1 1 0 0 ) + 0.2 ( 1 0 0 0 1 0 0 0 1 ) + 0.1 ( 0 1 0 1 0 0 0 0 1 ) + 0.5 ( 0 0 1 1 0 0 0 1 0 ) {\displaystyle 0.2{\begin{pmatrix}0&1&0\\0&0&1\\1&0&0\end{pmatrix}}+0.2{\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}+0.1{\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}}+0.5{\begin{pmatrix}0&0&1\\1&0&0\\0&1&0\end{pmatrix}}} Birkhoff's algorithm receives as input a bistochastic matrix and returns as output a Birkhoff decomposition. == Tools == A permutation set of an n-by-n matrix X is a set of n entries of X containing exactly one entry from each row and from each column. A theorem by Dénes Kőnig says that: Every bistochastic matrix has a permutation-set in which all entries are positive.The positivity graph of an n-by-n matrix X is a bipartite graph with 2n vertices, in which the vertices on one side are n rows and the vertices on the other side are the n columns, and there is an edge between a row and a column if the entry at that row and column is positive. A permutation set with positive entries is equivalent to a perfect matching in the positivity graph. A perfect matching in a bipartite graph can be found in polynomial time, e.g. using any algorithm for maximum cardinality matching. Kőnig's theorem is equivalent to the following:The positivity graph of any bistochastic matrix admits a perfect matching.A matrix is called scaled-bistochastic if all elements are non-negative, and the sum of each row and column equals c, where c is some positive constant. In other words, it is c times a bistochastic matrix. Since the positivity graph is not affected by scaling:The positivity graph of any scaled-bistochastic matrix admits a perfect matching. == Algorithm == Birkhoff's algorithm is a greedy algorithm: it greedily finds perfect matchings and removes them from the fractional matching. It works as follows. Let i = 1. Construct the positivity graph GX of X. Find a perfect matching in GX, corresponding to a positive permutation set in X. Let z[i] > 0 be the smallest entry in the permutation set. Let P[i] be a permutation matrix with 1 in the positive permutation set. Let X := X − z[i] P[i]. If X contains nonzero elements, Let i = i + 1 and go back to step 2. Otherwise, return the sum: z[1] P[1] + ... + z[2] P[2] + ... + z[i] P[i]. The algorithm is correct because, after step 6, the sum in each row and each column drops by z[i]. Therefore, the matrix X remains scaled-bistochastic. Therefore, in step 3, a perfect matching always exists. == Run-time complexity == By the selection of z[i] in step 4, in each iteration at least one element of X becomes 0. Therefore, the algorithm must end after at most n2 steps. However, the last step must simultaneously make n elements 0, so the algorithm ends after at most n2 − n + 1 steps, which implies O ( n 2 ) {\displaystyle O(n^{2})} . In 1960, Joshnson, Dulmage and Mendelsohn showed that Birkhoff's algorithm actually ends after at most n2 − 2n + 2 steps, which is tight in general (that is, in some cases n2 − 2n + 2 permutation matrices may be required). == Application in fair division == In the fair random assignment problem, there are n objects and n people with different preferences over the objects. It is required to give an object to each person. To attain fairness, the allocation is randomized: for each (person, object) pair, a probability is calculated, such that the sum of probabilities for each person and for each object is 1. The probabilistic-serial procedure can compute the probabilities such that each agent, looking at the matrix of probabilities, prefers his row of probabilities over the rows of all other people (this property is called envy-freeness). This raises the question of how to implement this randomized allocation in practice? One cannot just randomize for each object separately, since this may result in allocations in which some people get many objects while other people get no objects. Here, Birkhoff's algorithm is useful. The matrix of probabilities, calculated by the probabilistic-serial algorithm, is bistochastic. Birkhoff's algorithm can decompose it into a convex combination of permutation matrices. Each permutation matrix represents a deterministic assignment, in which every agent receives exactly one object. The coefficient of each such matrix is interpreted as a probability; based on the calculated probabilities, it is possible to pick one assignment at random and implement it. == Extensions == The problem of computing the Birkhoff decomposition with the minimum number of terms has been shown to be NP-hard, but some heuristics for computing it are known. This theorem can be extended for the general stochastic matrix with deterministic transition matrices. Budish, Che, Kojima and Milgrom generalize Birkhoff's algorithm to non-square matrices, with some constraints on the feasible assignments. They also present a decomposition algorithm that minimizes the variance in the expected values. Vazirani generalizes Birkhoff's algorithm to non-bipartite graphs. Valls et al. showed that it is possible to obtain an ϵ {\displaystyle \epsilon } -approximate decomposition with O ( log ⁡ ( 1 / ϵ 2 ) ) {\displaystyle O(\log(1/\epsilon ^{2}))} permutations.

Boyfriend Maker

Boyfriend Maker was a dating sim, romance chatbot smartphone app for iOS (iPhone) and Android devices, developed by Japanese studio 36 You Games (styled as 36You) and distributed under the freemium business model. Boyfriend Maker incorporated advanced artificial intelligence chat technology a decade before products such as ChatGPT. According to the developer's website, Boyfriend Maker is an "app that lets you interact and chat with quirky virtual boyfriends". While each virtual boyfriend has certain unique characteristics, the various instances of the boyfriend are powered by a chat engine, that (at least within a language and market) can utilise vocabulary and knowledge acquired in a chat with one user in subsequent chats with other users. == Gameplay == Users gain experience points and in-game coins. Users can customize their virtual boyfriend's appearance by selecting items such as hair, clothing, face, and a necklace. == Apple delisting and reintroduction == In late November 2012, the original iOS Boyfriend Maker app was delisted from the Apple App Store due to "ribald" chat, according to the New York Times. Boyfriend Maker was removed by Apple due to "reports of references to violent sexual acts and pedophilia". Boyfriend Maker had an age rating of 4+, even though the chat bot "responds with often strange and explicit text unsuitable for young children". User-posted chat excerpts indicate that the virtual boyfriend would sometimes transition abruptly to sexual chat in response to a seemingly innocent question. In one user-posted example, in response to the question, "what kind of wedding cake will we have" the boyfriend responds, "a good sex ima be on top of u u gonna ride oon me bitin the pillow gurrl ima fuck da shit out of u". The developer's use of the SimSimi-created third-party chat engine may be responsible for the sexual text. As the virtual boyfriend converses with human users, the SimSimi chat engine acquires vocabulary from users of the game and applies this "learned" vocabulary in chats with other users. The chat engine might also employ lines harvested from human-human chat logs, song lyrics, movies or TV shows. In April 2013, a detuned and presumably tamer version of the app, titled Boyfriend Plus, was permitted on Apple's App Store.

Top 10 AI Humanizers Compared (2026)

Looking for the best AI humanizer? An AI humanizer is software that uses machine learning to help you get more done — it can save you hours every week by automating repetitive work. Most options offer a generous free tier, with paid plans unlocking higher limits, faster processing, and team features. Whether you are a beginner or a pro, the right AI humanizer slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

Ashutosh Saxena

Ashutosh Saxena is an Indian-American computer scientist, researcher, and entrepreneur known for his contributions to the field of artificial intelligence and large-scale robot learning. His interests include building enterprise AI agents and embodied AI. Saxena is the co-founder and CEO of Caspar.AI, where generative AI parses data from ambient 3D radar sensors to predict 20+ health & wellness markers for pro-active patient care. Prior to Caspar.AI, Ashutosh co-founded Cognical Katapult (NSDQ: KPLT), which provides a no credit required alternative to traditional financing for online and omni-channel retail. Before Katapult, Saxena was an assistant professor in the Computer Science Department and faculty director of the RoboBrain Project (a large-scale AI model for robotics) at Cornell University. == Education == In 2009, with artificial intelligence pioneer Andrew Ng as his advisor, Saxena received both his M.S. and Ph.D. in computer science with an emphasis on artificial intelligence from Stanford University. Saxena received his bachelor's degree in electrical engineering from the Indian Institute of Technology, Kanpur in 2004. == Career == Saxena was the chief scientist of New York-based Holopad, where he worked with Steven Spielberg's team to create walkthroughs and 3D experiences for his movie TinTin. His past experiences include building acoustic AI models at Bose Corporation. Once Ashutosh completed his undergraduate degree, he became a researcher at the Commonwealth Scientific and Industrial Research Organization, where he developed AI models for medical devices. Before Caspar, Saxena pursued other entrepreneurial ventures, such as ZunaVision, an artificial intelligence startup he co-founded with Andrew Ng that uses AI to embed advertising space within videos. Ashutosh served as the CTO of ZunaVision from 2008 to 2010. After ZunaVision, Saxena co-founded Cognical Katapult, which provided financing solutions to nonprime and underbanked consumers powered by artificial intelligence. From 2014 to 2016, Saxena served as the faculty director of the RoboBrain project, which was a joint venture that he started between Stanford University, Cornell University, Brown University, and the University of California, Berkeley that made a knowledge engine for robots. Saxena co-founded Brain of Things in 2015 with David Cheriton, who serves as chief scientist, and was listed as the fastest growing private company reaching an annual recurring revenue of $8 million in three years. It has been widely covered in several outlets including Forbes Japan, and MIT Technology Review. Saxena's work on deep learning won test of time award in 2023 by Robotics Science and Systems. Ashutosh has been recognized for his work by receiving the Alfred P. Sloan Fellow in 2011, Google Faculty Research Award in 2012, Microsoft Faculty Fellowship in 2012, NSF Career award in 2013, One of the Eight Innovators to Watch by the Smithsonian Institution in 2015, and received TR35 Innovator Award by MIT Technology Review in 2018. He was named by San Francisco Business Times as a 40 under 40 young business leader. == Research == Saxena has authored over 100 published papers in the areas of large-scale robot learning and artificial intelligence, with 20,000+ citations. His work in the fields of computer vision and deep learning have been featured in press releases and academic journal reviews. Ashutosh's early work includes the Stanford Artificial Intelligence Robot (STAIR), an AI models that enables to perform tasks such as unload items from a dishwasher, which was covered on the front-page of New York Times. His work on Make3D, was the first work that estimated 3D depth from a single still image. At Cornell University, Ashutosh led the Robot Learning Lab, which used a machine learning approach to train robots to perform tasks in human environments such as generalizing manipulation in 3D point-clouds where robots learn to transfer manipulation trajectories to novel objects utilizing a large sample of demonstrations from crowdsourcing.

Top 10 AI Customer-support Bots Compared (2026)

Trying to pick the best AI customer-support bot? An AI customer-support bot is software that uses machine learning to help you get more done — it scales effortlessly from a single task to thousands. The best picks balance beginner-friendly simplicity with the depth power users need, and they ship updates often. Whether you are a beginner or a pro, the right AI customer-support bot slots into your workflow and pays for itself fast. Read on for hands-on impressions, pricing tiers, and the standout features that matter.

Confusion matrix

In machine learning, a confusion matrix, also known as error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a supervised learning one. In unsupervised learning it is usually called a matching matrix. The term is used specifically in the problem of statistical classification. Each row of the matrix represents the instances in an actual class while each column represents the instances in a predicted class, or vice versa – both variants are found in the literature. The diagonal of the matrix therefore represents all instances that are correctly predicted. The name stems from the fact that it makes it easy to identify whether the system is confusing two classes (i.e., commonly mislabeling one class as another). The confusion matrix has its origins in human perceptual studies of auditory stimuli. It was adapted for machine learning studies and used by Frank Rosenblatt, among other early researchers, to compare human and machine classifications of visual (and later auditory) stimuli. It is a special kind of contingency table, with two dimensions ("actual" and "predicted"), and identical sets of "classes" in both dimensions (each combination of dimension and class is a variable in the contingency table). == Example == Given a sample of 12 individuals, 8 that have been diagnosed with cancer and 4 that are cancer-free, where individuals with cancer belong to class 1 (positive) and non-cancer individuals belong to class 0 (negative), we can display that data as follows: Assume that we have a classifier that distinguishes between individuals with and without cancer in some way, we can take the 12 individuals and run them through the classifier. The classifier then makes 9 accurate predictions and misses 3: 2 individuals with cancer wrongly predicted as being cancer-free (sample 1 and 2), and 1 person without cancer that is wrongly predicted to have cancer (sample 9). Notice, that if we compare the actual classification set to the predicted classification set, there are 4 different outcomes that could result in any particular column: The actual classification is positive and the predicted classification is positive (1,1). This is called a true positive result because the positive sample was correctly identified by the classifier. The actual classification is positive and the predicted classification is negative (1,0). This is called a false negative result because the positive sample is incorrectly identified by the classifier as being negative. The actual classification is negative and the predicted classification is positive (0,1). This is called a false positive result because the negative sample is incorrectly identified by the classifier as being positive. The actual classification is negative and the predicted classification is negative (0,0). This is called a true negative result because the negative sample gets correctly identified by the classifier. We can then perform the comparison between actual and predicted classifications and add this information to the table, making correct results appear in green so they are more easily identifiable. The template for any binary confusion matrix uses the four kinds of results discussed above (true positives, false negatives, false positives, and true negatives) along with the positive and negative classifications. The four outcomes can be formulated in a 2×2 confusion matrix, as follows: The color convention of the three data tables above were picked to match this confusion matrix, in order to easily differentiate the data. Now, we can simply total up each type of result, substitute into the template, and create a confusion matrix that will concisely summarize the results of testing the classifier: In this confusion matrix, of the 8 samples with cancer, the system judged that 2 were cancer-free, and of the 4 samples without cancer, it predicted that 1 did have cancer. All correct predictions are located in the diagonal of the table (highlighted in green), so it is easy to visually inspect the table for prediction errors, as values outside the diagonal will represent them. By summing up the 2 rows of the confusion matrix, one can also deduce the total number of positive (P) and negative (N) samples in the original dataset, i.e. P = T P + F N {\displaystyle P=TP+FN} and N = F P + T N {\displaystyle N=FP+TN} . == Table of confusion == In predictive analytics, a table of confusion (sometimes also called a confusion matrix) is a table with two rows and two columns that reports the number of true positives, false negatives, false positives, and true negatives. This allows more detailed analysis than simply observing the proportion of correct classifications (accuracy). Accuracy will yield misleading results if the data set is unbalanced; that is, when the numbers of observations in different classes vary greatly. For example, if there were 95 cancer samples and only 5 non-cancer samples in the data, a particular classifier might classify all the observations as having cancer. The overall accuracy would be 95%, but in more detail the classifier would have a 100% recognition rate (sensitivity) for the cancer class but a 0% recognition rate for the non-cancer class. F1 score is even more unreliable in such cases, and here would yield over 97.4%, whereas informedness removes such bias and yields 0 as the probability of an informed decision for any form of guessing (here always guessing cancer). According to Davide Chicco and Giuseppe Jurman, the most informative metric to evaluate a confusion matrix is the Matthews correlation coefficient (MCC). Other metrics can be included in a confusion matrix, each of them having their significance and use. Some researchers have argued that the confusion matrix, and the metrics derived from it, do not truly reflect a model's knowledge. In particular, the confusion matrix cannot show whether correct predictions were reached through sound reasoning or merely by chance (a problem known in philosophy as epistemic luck). It also does not capture situations where the facts used to make a prediction later change or turn out to be wrong (defeasibility). This means that while the confusion matrix is a useful tool for measuring classification performance, it may give an incomplete picture of a model’s true reliability. == Confusion matrices with more than two categories == Confusion matrix is not limited to binary classification and can be used in multi-class classifiers as well. The confusion matrices discussed above have only two conditions: positive and negative. For example, the table below summarizes communication of a whistled language between two speakers, with zero values omitted for clarity. == Confusion matrices in multi-label and soft-label classification == Confusion matrices are not limited to single-label classification (where only one class is present) or hard-label settings (where classes are either fully present, 1, or absent, 0). They can also be extended to Multi-label classification (where multiple classes can be predicted at once) and soft-label classification (where classes can be partially present). One such extension is the Transport-based Confusion Matrix (TCM), which builds on the theory of optimal transport and the principle of maximum entropy. TCM applies to single-label, multi-label, and soft-label settings. It retains the familiar structure of the standard confusion matrix: a square matrix sized by the number of classes, with diagonal entries indicating correct predictions and off-diagonal entries indicating confusion. In the single-label case, TCM is identical to the standard confusion matrix. TCM follows the same reasoning as the standard confusion matrix: if class A is overestimated (its predicted value is greater than its label value) and class B is underestimated (its predicted value is less than its label value), A is considered confused with B, and the entry (B, A) is increased. If a class is both predicted and present, it is correctly identified, and the diagonal entry (A, A) increases. Optimal transport and maximum entropy are used to determine the extent to which these entries are updated. TCM enables clearer comparison between predictions and labels in complex classification tasks, while maintaining a consistent matrix format across settings.

ISO 2033

The ISO 2033:1983 standard ("Coding of machine readable characters (MICR and OCR)") defines character sets for use with Optical Character Recognition or Magnetic Ink Character Recognition systems. The Japanese standard JIS X 9010:1984 ("Coding of machine readable characters (OCR and MICR)", originally designated JIS C 6229-1984) is closely related. == Character set for OCR-A == The version of the encoding for the OCR-A font registered with the ISO-IR registry as ISO-IR-91 is the Japanese (JIS X 9010 / JIS C 6229) version, which differs from the encoding defined by ISO 2033 only in the addition of a Yen sign at 5C. == Character set for OCR-B == The version of the G0 set for the OCR-B font registered with the ISO-IR registry as ISO-IR-92 is the Japanese (JIS X 9010 / JIS C 6229) version, which differs from the encoding defined by ISO 2033 only in being based on JIS-Roman (with a dollar sign at 0x24 and a Yen sign at 0x5C) rather than on the ISO 646 IRV (with a backslash at 0x5C and, at the time, a universal currency sign (¤) at 0x24). Besides those code points, it differs from ASCII only in omitting the backtick (`) and tilde (~). An additional supplementary set registered as ISO-IR-93 assigns the pound sign (£), universal currency sign (¤) and section sign (§) to their ISO-8859-1 codepoints, and the backslash to the ISO-8859-1 codepoint for the Yen sign. == Character set for JIS X 9008 (JIS C 6257) == JIS X 9010 (JIS C 6229) also defines character sets for the JIS X 9008:1981 (formerly JIS C 6257-1981) "hand-printed" OCR font. These include subsets of the JIS X 0201 Roman set (registered as ISO-IR-94 and omitting the backtick (`), lowercase letters, curly braces ({, }) and overline (‾)), and kana set (registered as ISO-IR-96 and omitting the East Asian style comma (､) and full stop (｡), the interpunct (･) and the small kana), in addition to a set (registered as ISO-IR-95) containing only the backslash, which is assigned to the same code point as in ISO-IR-93. The JIS C 6527 font stylises the slash and backslash characters with a doubled appearance. The character names given are "Solidus" and "Reverse Solidus", matching the Unicode character names for the ASCII slash and backslash. However, the Unicode Optical Character Recognition block includes an additional code point for an "OCR Double Backslash" (⑊), although not for a double (forward) slash, although a double slash is available elsewhere, as U+2AFD ⫽ DOUBLE SOLIDUS OPERATOR. == Character set for E-13B == The ISO-IR-98 encoding defined by ISO 2033 encodes the character repertoire of the E13B font, as used with magnetic ink character recognition. Although ISO 2033 also specifies other encodings, the encoding for E-13B is the encoding referred to as ISO_2033_1983 by Perl libintl, and as ISO_2033-1983 or csISO2033 by the IANA. Other registered labels include iso-ir-98, its ISO-IR registration number, and simply e13b. The digits are preserved in their ASCII locations. Letters and symbols unavailable in the E13B font are omitted, while specialised punctuation for bank cheques included in the E13B font is added. The same symbols are available in Unicode in the Optical Character Recognition block.