The term evolvability is a framework of computational learning introduced by Leslie Valiant in his paper of the same name. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries. == General framework == Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely approximates f {\displaystyle f\,} . This closeness is measured by the performance Perf ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} of r {\displaystyle r\,} with respect to f {\displaystyle f\,} . As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some r , r ′ ∈ R n {\displaystyle r,r'\in R_{n}} , with r ≠ r ′ {\displaystyle r\neq r'\,} , still r ( x ) = r ′ ( x ) {\displaystyle r(x)=r'(x)\,} for all x ∈ X n {\displaystyle x\in X_{n}} . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the set of possible mutations of r {\displaystyle r\,} . For simplicity, consider Boolean functions on X n = { − 1 , 1 } n {\displaystyle X_{n}=\{-1,1\}^{n}\,} , and let D n {\displaystyle D_{n}\,} be a probability distribution on X n {\displaystyle X_{n}\,} . Define the performance in terms of this. Specifically, Perf ( f , r ) = ∑ x ∈ X n f ( x ) r ( x ) D n ( x ) . {\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ( f , r ) = Prob ( f ( x ) = r ( x ) ) − Prob ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship. Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by Perf s ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\displaystyle D_{n}\,} . If s {\displaystyle s\,} is large enough, evidently Perf s ( f , r ) {\displaystyle \operatorname {Perf} _{s}(f,r)} will be close to the actual performance Perf ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} . Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , initial representation r ∈ R n {\displaystyle r\in R_{n}} , sample size s {\displaystyle s\,} , and tolerance t {\displaystyle t\,} , the mutator Mut ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is a random variable defined as follows. Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ( f , r ′ ) − Perf s ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)\geq t} ; r ′ {\displaystyle r'\,} is a neutral mutation if − t < Perf s ( f , r ′ ) − Perf s ( f , r ) < t {\displaystyle -t<\operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)
Cyclodisparity
In vision science, cyclodisparity is the difference in the rotation angle of an object or scene viewed by the left and right eyes. Cyclodisparity can result from the eyes' torsional rotation (cyclorotation) or can be created artificially by presenting to the eyes two images that need to be rotated relative to each other for binocular fusion to take place. == Human and animal vision == The eyes and visual system can compensate for cyclodisparity up to a certain point; if the cyclodisparity is larger than a threshold, the images cannot be fused, resulting stereoblindness, and in double vision in subjects who otherwise have full stereo vision. When a human subject is presented with images that have artificial cyclodisparity, cyclovergence is evoked, that is, a motor response of the eye muscles that rotates the two eyes in opposite directions, thereby reducing cyclodisparity. Visually-induced cyclovergence of up to 8 degrees has been observed in normal subjects. Furthermore, up to about 8 degrees can usually be compensated by purely sensory means, that is, without physical eye rotation. This means that the normal human observer can achieve binocular image fusion in presence of cyclodisparity of up to approximately 16 degrees. Cyclodisparity due to images having been rotated inward can be compensated better when the gaze is directed downwards, and cyclodisparity due to an outward rotation can be compensated better when the gaze is directed upwards. A proposed explanation for this phenomenon is that the motor system is coordinated in such a way that the eyes perform a torsional movement to reduce the size of the search zones and thus the computational load required for solving the correspondence problem. The resulting cyclovergence at near gaze is smaller than the cyclovergence predicted by Listing's law. == Video processing and computer vision == Active camera torsion can be used in machine and computer vision for several purposes. For instance, camera torsion can be used to make improved use of the search range over which matching detectors or stereo matching algorithms operate, or to make a 3D slanted surface appear frontoparallel for further stereo processing. For image compression purposes, images with cyclodisparity are advantageously encoded using global motion compensation using a rotational motion model.
Pietro Perona
Pietro Perona (born 3 September 1961) is an Italian-American educator and computer scientist. He is the Allan E. Puckett Professor of Electrical Engineering and Computation and Neural Systems at the California Institute of Technology and director of the National Science Foundation Engineering Research Center in Neuromorphic Systems Engineering. He is known for his research in computer vision and is the director of the Caltech Computational Vision Group. == Academic biography == Perona obtained his D.Eng. in electrical engineering cum laude from the University of Padua in 1985 and completed his Ph.D. at the University of California, Berkeley in 1990. His dissertation was titled Finding Texture and Brightness Boundaries in Images, and his adviser was Jitendra Malik. In 1990, Perona was a postdoctoral fellow at the International Computer Science Institute at Berkeley. From 1990 to 1991, he was a postdoctoral fellow at the Massachusetts Institute of Technology in the Laboratory for Information and Decision Systems. He has been on the faculty of the California Institute of Technology since 1991, and he was named Allan E. Puckett Professor in 2008. == Research == Perona’s research focuses on the computational aspects of vision and learning. He developed the anisotropic diffusion equation, a partial differential equation that reduces noise in images while enhancing region boundaries. He is currently interested in visual recognition and in visual analysis of behavior. Perona and Serge Belongie lead the Visipedia project, which facilitates research on visual knowledge representation, visual search, and human-in-the-loop machine learning systems. Perona pioneered the study of visual categorization (including the publication of the Caltech 101 dataset) for which he was awarded the Longuet-Higgins Prize in 2013. He is also the recipient of the 2010 Koenderink Prize for Fundamental Contributions in Computer Vision, the 2003 Conference on Computer Vision and Pattern Recognition best paper award, and a 1996 NSF Presidential Young Investigator Award. == Media coverage == Perona has been quoted or had his research featured in various national media outlets, including the New York Times, Science Friday, The New Yorker, and the Los Angeles Times. In 2003, Perona and Stephen Nowlin organized the NEURO art exhibition, which brought together contemporary artists and scientists to explore neuromorphic engineering.
Nondeterministic finite automaton
In automata theory, a finite-state machine is called a deterministic finite automaton (DFA), if each of its transitions is uniquely determined by its source state and input symbol, and reading an input symbol is required for each state transition. A nondeterministic finite automaton (NFA), or nondeterministic finite-state machine, does not need to obey these restrictions. In particular, every DFA is also an NFA. Sometimes the term NFA is used in a narrower sense, referring to an NFA that is not a DFA, but not in this article. Using the subset construction algorithm, each NFA can be translated to an equivalent DFA; i.e., a DFA recognizing the same formal language. Like DFAs, NFAs only recognize regular languages. NFAs were introduced in 1959 by Michael O. Rabin and Dana Scott, who also showed their equivalence to DFAs. NFAs are used in the implementation of regular expressions: Thompson's construction is an algorithm for compiling a regular expression to an NFA that can efficiently perform pattern matching on strings. Conversely, Kleene's algorithm can be used to convert an NFA into a regular expression (whose size is generally exponential in the input automaton). NFAs have been generalized in multiple ways, e.g., nondeterministic finite automata with ε-moves, finite-state transducers, pushdown automata, alternating automata, ω-automata, and probabilistic automata. Besides the DFAs, other known special cases of NFAs are unambiguous finite automata (UFA) and self-verifying finite automata (SVFA). == Informal introduction == There are at least two equivalent ways to describe the behavior of an NFA. The first way makes use of the nondeterminism in the name of an NFA. For each input symbol, the NFA transitions to a new state until all input symbols have been consumed. In each step, the automaton nondeterministically "chooses" one of the applicable transitions. If there exists at least one "lucky run", i.e. some sequence of choices leading to an accepting state after completely consuming the input, it is accepted. Otherwise, i.e. if no choice sequence at all can consume all the input and lead to an accepting state, the input is rejected. In the second way, the NFA consumes a string of input symbols, one by one. In each step, whenever two or more transitions are applicable, it "clones" itself into appropriately many copies, each one following a different transition. If no transition is applicable, the current copy is in a dead end, and it "dies". If, after consuming the complete input, any of the copies is in an accept state, the input is accepted, else, it is rejected. == Formal definition == For a more elementary introduction of the formal definition, see automata theory. === Automaton === An NFA is represented formally by a 5-tuple, ( Q , Σ , δ , q 0 , F ) {\displaystyle (Q,\Sigma ,\delta ,q_{0},F)} , consisting of a finite set of states Q {\displaystyle Q} , a finite set of input symbols called the alphabet Σ {\displaystyle \Sigma } , a transition function δ {\displaystyle \delta } : Q × Σ → P ( Q ) {\displaystyle Q\times \Sigma \rightarrow {\mathcal {P}}(Q)} , an initial (or start) state q 0 ∈ Q {\displaystyle q_{0}\in Q} , and a set of accepting (or final) states F ⊆ Q {\displaystyle F\subseteq Q} . Here, P ( Q ) {\displaystyle {\mathcal {P}}(Q)} denotes the power set of Q {\displaystyle Q} . === Recognized language === Given an NFA M = ( Q , Σ , δ , q 0 , F ) {\displaystyle M=(Q,\Sigma ,\delta ,q_{0},F)} , its recognized language is denoted by L ( M ) {\displaystyle L(M)} , and is defined as the set of all strings over the alphabet Σ {\displaystyle \Sigma } that are accepted by M {\displaystyle M} . Loosely corresponding to the above informal explanations, there are several equivalent formal definitions of a string w = a 1 a 2 . . . a n {\displaystyle w=a_{1}a_{2}...a_{n}} being accepted by M {\displaystyle M} : w {\displaystyle w} is accepted if a sequence of states, r 0 , r 1 , . . . , r n {\displaystyle r_{0},r_{1},...,r_{n}} , exists in Q {\displaystyle Q} such that: r 0 = q 0 {\displaystyle r_{0}=q_{0}} r i + 1 ∈ δ ( r i , a i + 1 ) {\displaystyle r_{i+1}\in \delta (r_{i},a_{i+1})} , for i = 0 , … , n − 1 {\displaystyle i=0,\ldots ,n-1} r n ∈ F {\displaystyle r_{n}\in F} . In words, the first condition says that the machine starts in the start state q 0 {\displaystyle q_{0}} . The second condition says that given each character of string w {\displaystyle w} , the machine will transition from state to state according to the transition function δ {\displaystyle \delta } . The last condition says that the machine accepts w {\displaystyle w} if the last input of w {\displaystyle w} causes the machine to halt in one of the accepting states. In order for w {\displaystyle w} to be accepted by M {\displaystyle M} , it is not required that every state sequence ends in an accepting state, it is sufficient if one does. Otherwise, i.e. if it is impossible at all to get from q 0 {\displaystyle q_{0}} to a state from F {\displaystyle F} by following w {\displaystyle w} , it is said that the automaton rejects the string. The set of strings M {\displaystyle M} accepts is the language recognized by M {\displaystyle M} and this language is denoted by L ( M ) {\displaystyle L(M)} . Alternatively, w {\displaystyle w} is accepted if δ ∗ ( q 0 , w ) ∩ F ≠ ∅ {\displaystyle \delta ^{}(q_{0},w)\cap F\not =\emptyset } , where δ ∗ : Q × Σ ∗ → P ( Q ) {\displaystyle \delta ^{}:Q\times \Sigma ^{}\rightarrow {\mathcal {P}}(Q)} is defined recursively by: δ ∗ ( r , ε ) = { r } {\displaystyle \delta ^{}(r,\varepsilon )=\{r\}} where ε {\displaystyle \varepsilon } is the empty string, and δ ∗ ( r , x a ) = ⋃ r ′ ∈ δ ∗ ( r , x ) δ ( r ′ , a ) {\displaystyle \delta ^{}(r,xa)=\bigcup _{r'\in \delta ^{}(r,x)}\delta (r',a)} for all x ∈ Σ ∗ , a ∈ Σ {\displaystyle x\in \Sigma ^{},a\in \Sigma } . In words, δ ∗ ( r , x ) {\displaystyle \delta ^{}(r,x)} is the set of all states reachable from state r {\displaystyle r} by consuming the string x {\displaystyle x} . The string w {\displaystyle w} is accepted if some accepting state in F {\displaystyle F} can be reached from the start state q 0 {\displaystyle q_{0}} by consuming w {\displaystyle w} . === Initial state === The above automaton definition uses a single initial state, which is not necessary. Sometimes, NFAs are defined with a set of initial states. There is an easy construction that translates an NFA with multiple initial states to an NFA with a single initial state, which provides a convenient notation. == Example == The following automaton M, with a binary alphabet, determines if the input ends with a 1. Let M = ( { p , q } , { 0 , 1 } , δ , p , { q } ) {\displaystyle M=(\{p,q\},\{0,1\},\delta ,p,\{q\})} where the transition function δ {\displaystyle \delta } can be defined by this state transition table (cf. upper left picture): State Input 0 1 p { p } { p , q } q ∅ ∅ {\displaystyle {\begin{array}{|c|cc|}{\bcancel {{}_{\text{State}}\quad {}^{\text{Input}}}}&0&1\\\hline p&\{p\}&\{p,q\}\\q&\emptyset &\emptyset \end{array}}} Since the set δ ( p , 1 ) {\displaystyle \delta (p,1)} contains more than one state, M is nondeterministic. The language of M can be described by the regular language given by the regular expression (0|1)1. All possible state sequences for the input string "1011" are shown in the lower picture. The string is accepted by M since one state sequence satisfies the above definition; it does not matter that other sequences fail to do so. The picture can be interpreted in a couple of ways: In terms of the above "lucky-run" explanation, each path in the picture denotes a sequence of choices of M. In terms of the "cloning" explanation, each vertical column shows all clones of M at a given point in time, multiple arrows emanating from a node indicate cloning, a node without emanating arrows indicating the "death" of a clone. The feasibility to read the same picture in two ways also indicates the equivalence of both above explanations. Considering the first of the above formal definitions, "1011" is accepted since when reading it M may traverse the state sequence ⟨ r 0 , r 1 , r 2 , r 3 , r 4 ⟩ = ⟨ p , p , p , p , q ⟩ {\displaystyle \langle r_{0},r_{1},r_{2},r_{3},r_{4}\rangle =\langle p,p,p,p,q\rangle } , which satisfies conditions 1 to 3. Concerning the second formal definition, bottom-up computation shows that δ ∗ ( p , ε ) = { p } {\displaystyle \delta ^{}(p,\varepsilon )=\{p\}} , hence δ ∗ ( p , 1 ) = δ ( p , 1 ) = { p , q } {\displaystyle \delta ^{}(p,1)=\delta (p,1)=\{p,q\}} , hence δ ∗ ( p , 10 ) = δ ( p , 0 ) ∪ δ ( q , 0 ) = { p } ∪ { } {\displaystyle \delta ^{}(p,10)=\delta (p,0)\cup \delta (q,0)=\{p\}\cup \{\}} , hence δ ∗ ( p , 101 ) = δ ( p , 1 ) = { p , q } {\displaystyle \delta ^{}(p,101)=\delta (p,1)=\{p,q\}} , and hence δ ∗ ( p , 1011 ) = δ ( p , 1 ) ∪ δ ( q , 1 ) = { p , q } ∪ { } {\displaystyle \delta ^{}(p,1011)=\delta (p,1)\cup \delta (q,1)=\{p,q\}\cup \{\}} ; since that set is
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Stability (learning theory)
Stability, also known as algorithmic stability, is a notion in computational learning theory of how a machine learning algorithm output is changed with small perturbations to its inputs. A stable learning algorithm is one for which the prediction does not change much when the training data is modified slightly. For instance, consider a machine learning algorithm that is being trained to recognize handwritten letters of the alphabet, using 1000 examples of handwritten letters and their labels ("A" to "Z") as a training set. One way to modify this training set is to leave out an example, so that only 999 examples of handwritten letters and their labels are available. A stable learning algorithm would produce a similar classifier with both the 1000-element and 999-element training sets. Stability can be studied for many types of learning problems, from language learning to inverse problems in physics and engineering, as it is a property of the learning process rather than the type of information being learned. The study of stability gained importance in computational learning theory in the 2000s when it was shown to have a connection with generalization. It was shown that for large classes of learning algorithms, notably empirical risk minimization algorithms, certain types of stability ensure good generalization. == History == A central goal in designing a machine learning system is to guarantee that the learning algorithm will generalize, or perform accurately on new examples after being trained on a finite number of them. In the 1990s, milestones were reached in obtaining generalization bounds for supervised learning algorithms. The technique historically used to prove generalization was to show that an algorithm was consistent, using the uniform convergence properties of empirical quantities to their means. This technique was used to obtain generalization bounds for the large class of empirical risk minimization (ERM) algorithms. An ERM algorithm is one that selects a solution from a hypothesis space H {\displaystyle H} in such a way to minimize the empirical error on a training set S {\displaystyle S} . A general result, proved by Vladimir Vapnik for an ERM binary classification algorithms, is that for any target function and input distribution, any hypothesis space H {\displaystyle H} with VC-dimension d {\displaystyle d} , and n {\displaystyle n} training examples, the algorithm is consistent and will produce a training error that is at most O ( d n ) {\displaystyle O\left({\sqrt {\frac {d}{n}}}\right)} (plus logarithmic factors) from the true error. The result was later extended to almost-ERM algorithms with function classes that do not have unique minimizers. Vapnik's work, using what became known as VC theory, established a relationship between generalization of a learning algorithm and properties of the hypothesis space H {\displaystyle H} of functions being learned. However, these results could not be applied to algorithms with hypothesis spaces of unbounded VC-dimension. Put another way, these results could not be applied when the information being learned had a complexity that was too large to measure. Some of the simplest machine learning algorithms—for instance, for regression—have hypothesis spaces with unbounded VC-dimension. Another example is language learning algorithms that can produce sentences of arbitrary length. Stability analysis was developed in the 2000s for computational learning theory and is an alternative method for obtaining generalization bounds. The stability of an algorithm is a property of the learning process, rather than a direct property of the hypothesis space H {\displaystyle H} , and it can be assessed in algorithms that have hypothesis spaces with unbounded or undefined VC-dimension such as nearest neighbor. A stable learning algorithm is one for which the learned function does not change much when the training set is slightly modified, for instance by leaving out an example. A measure of Leave one out error is used in a Cross Validation Leave One Out (CVloo) algorithm to evaluate a learning algorithm's stability with respect to the loss function. As such, stability analysis is the application of sensitivity analysis to machine learning. == Summary of classic results == Early 1900s - Stability in learning theory was earliest described in terms of continuity of the learning map L {\displaystyle L} , traced to Andrey Nikolayevich Tikhonov. 1979 - Devroye and Wagner observed that the leave-one-out behavior of an algorithm is related to its sensitivity to small changes in the sample. 1999 - Kearns and Ron discovered a connection between finite VC-dimension and stability. 2002 - In a landmark paper, Bousquet and Elisseeff proposed the notion of uniform hypothesis stability of a learning algorithm and showed that it implies low generalization error. Uniform hypothesis stability, however, is a strong condition that does not apply to large classes of algorithms, including ERM algorithms with a hypothesis space of only two functions. 2002 - Kutin and Niyogi extended Bousquet and Elisseeff's results by providing generalization bounds for several weaker forms of stability which they called almost-everywhere stability. Furthermore, they took an initial step in establishing the relationship between stability and consistency in ERM algorithms in the Probably Approximately Correct (PAC) setting. 2004 - Poggio et al. proved a general relationship between stability and ERM consistency. They proposed a statistical form of leave-one-out-stability which they called CVEEEloo stability, and showed that it is a) sufficient for generalization in bounded loss classes, and b) necessary and sufficient for consistency (and thus generalization) of ERM algorithms for certain loss functions such as the square loss, the absolute value and the binary classification loss. 2010 - Shalev Shwartz et al. noticed problems with the original results of Vapnik due to the complex relations between hypothesis space and loss class. They discuss stability notions that capture different loss classes and different types of learning, supervised and unsupervised. 2016 - Moritz Hardt et al. proved stability of gradient descent given certain assumption on the hypothesis and number of times each instance is used to update the model. == Preliminary definitions == We define several terms related to learning algorithms training sets, so that we can then define stability in multiple ways and present theorems from the field. A machine learning algorithm, also known as a learning map L {\displaystyle L} , maps a training data set, which is a set of labeled examples ( x , y ) {\displaystyle (x,y)} , onto a function f {\displaystyle f} from X {\displaystyle X} to Y {\displaystyle Y} , where X {\displaystyle X} and Y {\displaystyle Y} are in the same space of the training examples. The functions f {\displaystyle f} are selected from a hypothesis space of functions called H {\displaystyle H} . The training set from which an algorithm learns is defined as S = { z 1 = ( x 1 , y 1 ) , . . , z m = ( x m , y m ) } {\displaystyle S=\{z_{1}=(x_{1},\ y_{1})\ ,..,\ z_{m}=(x_{m},\ y_{m})\}} and is of size m {\displaystyle m} in Z = X × Y {\displaystyle Z=X\times Y} drawn i.i.d. from an unknown distribution D. Thus, the learning map L {\displaystyle L} is defined as a mapping from Z m {\displaystyle Z_{m}} into H {\displaystyle H} , mapping a training set S {\displaystyle S} onto a function f S {\displaystyle f_{S}} from X {\displaystyle X} to Y {\displaystyle Y} . Here, we consider only deterministic algorithms where L {\displaystyle L} is symmetric with respect to S {\displaystyle S} , i.e. it does not depend on the order of the elements in the training set. Furthermore, we assume that all functions are measurable and all sets are countable. The loss V {\displaystyle V} of a hypothesis f {\displaystyle f} with respect to an example z = ( x , y ) {\displaystyle z=(x,y)} is then defined as V ( f , z ) = V ( f ( x ) , y ) {\displaystyle V(f,z)=V(f(x),y)} . The empirical error of f {\displaystyle f} is I S [ f ] = 1 n ∑ V ( f , z i ) {\displaystyle I_{S}[f]={\frac {1}{n}}\sum V(f,z_{i})} . The true error of f {\displaystyle f} is I [ f ] = E z V ( f , z ) {\displaystyle I[f]=\mathbb {E} _{z}V(f,z)} Given a training set S of size m, we will build, for all i = 1....,m, modified training sets as follows: By removing the i-th element S | i = { z 1 , . . . , z i − 1 , z i + 1 , . . . , z m } {\displaystyle S^{|i}=\{z_{1},...,\ z_{i-1},\ z_{i+1},...,\ z_{m}\}} By replacing the i-th element S i = { z 1 , . . . , z i − 1 , z i ′ , z i + 1 , . . . , z m } {\displaystyle S^{i}=\{z_{1},...,\ z_{i-1},\ z_{i}',\ z_{i+1},...,\ z_{m}\}} == Definitions of stability == === Hypothesis Stability === An algorithm L {\displaystyle L} has hypothesis stability β with respect to the loss function V if the following holds: ∀ i ∈ { 1 , . . . , m } , E S , z [ | V ( f S , z ) − V ( f S |
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