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Search problem in quantum mechanics
The hidden linear function problem, is a search problem that generalizes the Bernstein–Vazirani problem. In the Bernstein–Vazirani problem, the hidden function
Hidden linear function problem
Hidden_linear_function_problem
systems of linear equations Hidden linear function problem: oracle problem involving a hidden linear function Hidden shift problem: problem of finding
List_of_algorithms
Seven mathematical problems with a US$1 million prize for each solution
Millennium Prize Problems are seven well-known complex mathematical problems selected by the Clay Mathematics Institute in 2000. Each of the problems has a prize
Millennium_Prize_Problems
Type of activation function
(rectified linear unit) activation function is an activation function defined as the non-negative part of its argument, i.e., the ramp function: ReLU(x)=x+=max(0
Rectified_linear_unit
System where changes of output are not proportional to changes of input
(or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest
Nonlinear_system
Algorithm for supervised learning of binary classifiers
is a function that can decide whether or not an input, represented by a vector of numbers, belongs to some specific class. It is a type of linear classifier
Perceptron
nonlinear control problems. It is based on quasi-linearization, which is the approximation of the non-linear system under investigation by a linear time-invariant
Describing_function
Theoretical problem in quantum physics
definite result. The wave function in quantum mechanics evolves deterministically according to the Schrödinger equation as a linear superposition of different
Measurement_problem
Type of feedforward neural network
with nonlinear activation functions, organized in layers, notable for being able to distinguish data that is not linearly separable. Modern neural networks
Multilayer_perceptron
Problem in computer science
In quantum computing, the hidden shift problem is a type of oracle-based problem. Various versions of this problem have quantum algorithms which can run
Hidden_shift_problem
List of quantum computing algorithms
including algorithms, algorithmic techniques, computational models, and problem frameworks used in quantum computing. A quantum algorithm is an algorithm
List_of_quantum_algorithms
Estimate of time taken for running an algorithm
according to the type of function appearing in the big O notation. For example, an algorithm with time complexity O(n) is a linear time algorithm and an
Time_complexity
Artificial neural network node function
likely to suffer from the vanishing gradient problem. Ridge functions are multivariate functions acting on a linear combination of the input variables. Often
Activation_function
Very general problem in computer science
H. Hidden subgroup problem: Let G {\displaystyle G} be a group, X {\displaystyle X} a finite set, and f : G → X {\displaystyle f:G\to X} a function that
Hidden_subgroup_problem
Type of artificial neural network
function. Circa 1800, Legendre (1805) and Gauss (1795) created the simplest feedforward network which consists of a single weight layer with linear activation
Feedforward_neural_network
Statistical modeling method
variables is a multiple linear regression. In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters
Linear_regression
Type of artificial neural network
basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output
Radial_basis_function_network
Method of improving artificial neural network
Since the parameters of each hidden unit converge linearly, the whole optimization problem has a linear rate of convergence. Ioffe, Sergey; Szegedy
Batch_normalization
Smooth approximation of one-hot arg max
linear discriminant analysis, the input to the function is the result of K distinct linear functions, and the predicted probability for the j-th class
Softmax_function
Neural networks
introducing stronger non-linearities (either in the energy function or neurons’ activation functions) leading to super-linear (even an exponential) memory
Modern_Hopfield_network
Quantum algorithm
same problem. Therefore, Bernstein-Vazirani provides a super-polynomial separation between BPP and BQP. Hidden Linear Function problem Simon's problem Ethan
Bernstein–Vazirani_algorithm
Unsolved problem in computer science
Unsolved problem in computer science If the solution to a problem can be verified in polynomial time, must the problem be solvable in polynomial time
P_versus_NP_problem
Computation complexity problem
In quantum information, the hidden matching problem is a computational complexity problem that can be solved using quantum protocols: Let n {\displaystyle
Hidden_matching_problem
Problem in computer science
Simon's problem, usually called Simon's algorithm, served as the inspiration for Shor's algorithm. Both problems are special cases of the abelian hidden subgroup
Simon's_problem
Book by Marvin Minsky and Seymour Papert
any classification problem. (Existence theorem.) Minsky and Papert used perceptrons with restricted numbers of inputs of the hidden layer A-elements and
Perceptrons_(book)
Control theory for nonlinear or time-variant systems
treated as linear for purposes of control design: Feedback linearization And Lyapunov based methods: Lyapunov redesign Control-Lyapunov function Nonlinear
Nonlinear_control
Set of methods for supervised statistical learning
all i. This is called the dual problem. Since the dual maximization problem is a quadratic function of the ci subject to linear constraints, it is efficiently
Support_vector_machine
Public key cryptosystem
Hidden Fields Equations (HFE), also known as HFE trapdoor function, is a public key cryptosystem which was introduced at Eurocrypt in 1996 and proposed
Hidden_Field_Equations
Generalized function whose value is zero everywhere except at zero
developed the theory of distributions, where it is defined as a linear form acting on functions. The graph of the Dirac delta is usually thought of as following
Dirac_delta_function
Conjecture on zeros of the zeta function
Unsolved problem in mathematics Do all non-trivial zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics
Riemann_hypothesis
Hypersurface used by a classification algorithm
the number of hidden layers the network has. If it has no hidden layers, then it can only learn linear problems. If it has one hidden layer, then it
Decision_boundary
Classical problem in combinatorics
1} . This linear program belongs to the more general class of LPs for covering problems, as all the coefficients in the objective function and both sides
Set_cover_problem
Statistical Markov model
maximum likelihood estimation. For linear chain HMMs, the Baum–Welch algorithm can be used to estimate parameters. Hidden Markov models are known for their
Hidden_Markov_model
Aizerman problem states that a linear system in feedback with a sector nonlinearity would be stable if the linear system is stable for any linear gain of
Aizerman's_conjecture
Process by which a quantum system takes on a definitive state
E, z components of spin (sz), and so on. The observable acts as a linear function on the states of the system; its eigenvectors correspond to the quantum
Wave_function_collapse
Machine learning technique
according to the problem being solved: Experts f1,...,fn, each taking the same input x, and producing outputs f1(x),...,fn(x). A weighting function (also known
Mixture_of_experts
Of a function, an additional effect besides returning a value
Unspecified behaviour Frame problem Spuler, David A.; Sajeev, A. Sayed Muhammed (January 1994). Compiler Detection of Function Call Side Effects. James Cook
Side effect (computer science)
Side_effect_(computer_science)
On topology of algebraic curves and surfaces
regions of attraction, which are hidden attractors, and semi-stable limit cycles. In his speech, Hilbert presented the problems as: The upper bound of closed
Hilbert's_sixteenth_problem
Optimization algorithm for artificial neural networks
multi-class classification, while for the hidden layers this was traditionally a sigmoid function (logistic function or others) on each node (coordinate),
Backpropagation
Type of artificial neural network
cases, the output weights of hidden nodes are usually learned in a single step, which essentially amounts to learning a linear model. The name "extreme learning
Extreme_learning_machine
Machine learning model training problem
xt)=F(ht−1,ut,θ) Often, the output xt is a function of ht, as some xt=G(ht). The vanishing gradient problem already presents itself clearly when xt=ht
Vanishing_gradient_problem
Property of artificial neural networks
a network with a single hidden layer), using sigmoid activation functions in the hidden layer and linear activation functions in the input and output
Universal approximation theorem
Universal_approximation_theorem
Overview of and topical guide to algorithms
tree search Automated planning and scheduling Constraint satisfaction problem Linear regression Logistic regression Decision tree learning Random forest
Outline_of_algorithms
1932 book by John von Neumann
mathematical argument against the idea of hidden variables. Von Neumann's claim rested on the assumption that any linear combination of Hermitian operators represents
Mathematical Foundations of Quantum Mechanics
Mathematical_Foundations_of_Quantum_Mechanics
Family of machine learning approaches
0th hidden vector of decoder. Then, the intermediate vector is transformed by a linear map WQ into a query vector q0=h0dWQ. Meanwhile, the hidden vectors
Seq2seq
manifolds. In the category of piecewise linear manifolds, the generalized conjecture is still an open problem for 4-dimensional manifolds. For differential
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Regulator for flow of signals in neural networks
gates: a reset gate and an update gate. GRU also merges the cell state and hidden state. The reset gate roughly corresponds to the forget gate, and the update
Gating_mechanism
Algorithm to be run on quantum computers
S2CID 2337707. Boneh, D.; Lipton, R. J. (1995). "Quantum cryptoanalysis of hidden linear functions". In Coppersmith, D. (ed.). Proceedings of the 15th Annual International
Quantum_algorithm
Task of computing complete subgraphs
significantly less than linear. The clique decision problem is NP-complete. It was one of Richard Karp's original 21 problems shown NP-complete in his
Clique_problem
Optimization algorithm
a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method
Gradient_descent
Sequence of data points over time
a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a
Time_series
Statistical model for a binary dependent variable
classifier. Analogous linear models for binary variables with a different sigmoid function instead of the logistic function (to convert the linear combination to
Logistic_regression
Type of differential equation
PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function u of x and y, a second order linear PDE is of the
Partial_differential_equation
Machine learning technique
linear regression is equivalent to fitting complex parameters θ1,…,θD∈ℂ such thatfθ(x)=Re(∑kθkei⟨ωk,x⟩)which is a neural network with a single hidden
Random_feature
Conflict of interest when one person acts on another's behalf
typically either examine moral hazard (hidden actions) or adverse selection (hidden information). The principal–agent problem typically arises where the two parties
Principal–agent_problem
Function related to statistics and probability theory
reduces computational burden of the original maximization problem. For instance, in a linear regression with normally distributed errors, y = X β + u {\textstyle
Likelihood_function
Methods for numerical approximations
objective function and the constraint. For instance, linear programming deals with the case that both the objective function and the constraints are linear. A
Numerical_analysis
Evolutionary algorithm
the square root function. This kind of expression tree consists of the phenotypic expression of GEP genes, whereas the genes are linear strings encoding
Gene_expression_programming
Method for nonparametric multiple regression
smoothing functions to these explanatory variables. The model consists of linear combinations of ridge functions: non-linear transformations of linear combinations
Projection_pursuit_regression
Mathematical description of quantum state
advantages to understanding wave functions as representing elements of an abstract vector space: All the powerful tools of linear algebra can be used to manipulate
Wave_function
Supervised machine learning techniques
a function ϕ(x,y) that maps a training sample x and a candidate prediction y to a vector of length n (x and y may have any structure; n is problem-dependent
Structured_prediction
Frequency swept signal
rotational acceleration. Linear chirp Sound example for linear chirp (five repetitions) Problems playing this file? See media help. In a linear-frequency chirp
Chirp
unknown probability density function recursively over time using incremental incoming measurements. It is one of the main problems defined by Norbert Wiener
Smoothing problem (stochastic processes)
Smoothing_problem_(stochastic_processes)
algebra Stochastic satisfiability Linear temporal logic satisfiability and model checking Type inhabitation problem for simply typed lambda calculus Integer
List of PSPACE-complete problems
List_of_PSPACE-complete_problems
Interpretation of quantum mechanics
interpretation, and hidden variable theories such as Bohmian mechanics. In the many-worlds interpretation, the universal wave function evolves unitarily
Many-worlds_interpretation
Overview of and topical guide to machine learning
Layered hidden Markov model Learnable function class Least squares support vector machine Leslie P. Kaelbling Linear genetic programming Linear predictor
Outline_of_machine_learning
Property of a model
into the method. E.g., when approximating a non-linear function f(x) using a learning method for linear models, there will be error in the estimates f̂(x)
Bias–variance_tradeoff
Framework for machine learning
example would be restricting ℋ to linear functions: this can be seen as a reduction to the standard problem of linear regression. ℋ could also be restricted
Statistical_learning_theory
Statistical estimation technique
The model assumes that the conditional mean of 𝐲 given 𝐗 to be a linear function of 𝐗 and that the conditional variance of the error term given 𝐗
Generalized_least_squares
Projection of data onto lower-dimensional manifolds
potentially existing across non-linear manifolds (non-affine subspaces) which cannot be adequately captured by linear decomposition methods, onto lower-dimensional
Nonlinear dimensionality reduction
Nonlinear_dimensionality_reduction
Flaw in mathematical modelling
slopes). Replacing this simple function with a new, more complex quadratic function, or with a new, more complex linear function on more than two independent
Overfitting
Algorithm that estimates unknowns from a series of measurements over time
the linear-quadratic regulator (LQR), the Kalman filter solves the linear–quadratic–Gaussian control problem (LQG). The Kalman filter, the linear-quadratic
Kalman_filter
correction Best linear unbiased prediction Beta (finance) Beta-binomial distribution Beta-binomial model Beta distribution Beta function – for incomplete
List_of_statistics_articles
Optimization algorithm
statistical estimation and machine learning consider the problem of minimizing an objective function that has the form of a sum: Q(w)=1n∑i=1nQi(w), where
Stochastic_gradient_descent
Neural network that learns efficient data encoding in an unsupervised manner
to, the message space 𝒳, or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental
Autoencoder
Form of artificial neural network
introducing stronger non-linearities (either in the energy function or neurons' activation functions) leading to super-linear (even an exponential) memory
Hopfield_network
Class of algorithms for pattern analysis
support-vector machine (SVM). These methods involve using linear classifiers to solve nonlinear problems. The general task of pattern analysis is to find and
Kernel_method
Classification of Artificial Neural Networks (ANNs)
statistics. In classification problems the output layer is typically a sigmoid function of a linear combination of hidden layer values, representing a
Types of artificial neural networks
Types_of_artificial_neural_networks
Set of machine learning methods
parameter to the minimization problem of the learning algorithm. As an example, consider the case of supervised learning of a linear combination of a set of
Multiple_kernel_learning
numerical method for linear problems Nikolay Krylov, author of the edge-of-the-wedge theorem, Krylov–Bogolyubov theorem and describing function Aleksandr Kurosh
List of Russian mathematicians
List_of_Russian_mathematicians
Set of statistical processes for estimating the relationships among variables
Function approximation Generalized linear model Kriging (a linear least squares estimation algorithm) Local regression Modifiable areal unit problem Multivariate
Regression_analysis
Technique to solve partial differential equations
function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem
Physics-informed neural networks
Physics-informed_neural_networks
Iterative method for finding maximum likelihood estimates in statistical models
to estimate a mixture of gaussians, or to solve the multiple linear regression problem. The EM algorithm was explained and given its name in a classic
Expectation–maximization algorithm
Expectation–maximization_algorithm
Class of artificial neural network
weights in a neural network can be modeled as a non-linear global optimization problem. A target function can be formed to evaluate the fitness or error of
Recurrent_neural_network
linear stochastic partial differential equation for the un-normalized density of a hidden state. In contrast, the Kushner equation gives a non-linear
Zakai_equation
Branch of mathematics
2x is its derivative. If a function is linear (that is if the graph of the function is a straight line), then the function can be written as y = mx +
Calculus
Concept in machine learning
problem is to minimize expected loss (also known as the risk), defined as I[f]=∫𝒳×𝒴V(f(x→),y)p(x→,y)dx→dy where V(f(x→),y) is a given loss function
Loss functions for classification
Loss_functions_for_classification
Process of reducing the number of random variables under consideration
and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Linear approaches can be further divided into feature selection
Dimensionality_reduction
algorithm that can learn kernel machines, i.e. non-linear classifiers that employ a kernel function to compute the similarity of unseen samples to training
Kernel_perceptron
Class of second-order linear partial differential equations
matrix-valued function a ( x ) {\displaystyle a(x)} has a kernel of dimension 1. Under broad assumptions, an initial/boundary-value problem for a linear parabolic
Parabolic partial differential equation
Parabolic_partial_differential_equation
Signal processing computational method
another. For instance, two grayscale images can be linearly combined to create mixed images in which the hidden content is visually imperceptible. ICA can then
Independent component analysis
Independent_component_analysis
Quantum search algorithm
Umesh Vazirani proved that any quantum solution to the problem needs to evaluate the function Ω ( N ) {\displaystyle \Omega ({\sqrt {N}})} times, so Grover's
Grover's_algorithm
Machine learning framework
formulated as a sequence of alternating linear integral operators on function spaces and point-wise non-linearities. Using an analogous architecture to finite-dimensional
Neural_operators
Class of statistical modeling methods
probability of a certain value for Yi. Linear-chain CRFs have many of the same applications as conceptually simpler hidden Markov models (HMMs), but relax certain
Conditional_random_field
Hyperbolic analogues of trigonometric functions
x=\cosh x\,.} All functions with this property are linear combinations of sinh and cosh, in particular the exponential functions e x {\displaystyle e^{x}}
Hyperbolic_functions
Complete subgraph added to a random graph
achieve the same bound on k and runs in linear time. It is also possible to solve the planted clique problem, regardless of the choice of k, in quasi-polynomial
Planted_clique
Electronic circuit that behaves chaotically
containing a nonlinear element such that f(x) was a 3-segment piecewise-linear function. The easy experimental implementation of the circuit, combined with
Chua's_circuit
Method of machine learning
example, with other convex loss functions. Consider the setting of supervised learning with f being a linear function to be learned: f(xj)=⟨w,xj⟩=w⋅xj
Online_machine_learning
Attribute of machine learning models
function using a finite number of training samples. However, if we are only interested in a particular class of target functions (e.g., only linear functions)
Sample_complexity
Technique for the generative modeling of a continuous probability distribution
presents a problem for learning the score function, because if there are no samples around a certain point, then we can't learn the score function at that
Diffusion_model
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HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
HIDDEN LINEAR-FUNCTION-PROBLEM
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