One Piece of Math a Day

Convolution and Fourier

2024-11-06

Convolution theorem: Fourier transform of the convolution of two functions (under suitable assumptions) is the product of the Fourier transforms of these two functions. [ref]

Dual numbers

2024-11-05

Dual numbers correspond to the completion of the real line with an nilpotent element ε different from 0. It can be thought as a universal linearization of functions, or as the forward-mode of automatic differentiation. [ref]

Convergence of iterative differentiation

2024-11-04

Convergence of iterates does not imply convergence of the derivatives. Nevertheless, Gilbert (1994) proposed an interversion limit-derivative theorem under strong assumption on the spectrum of the derivatives. [ref]

Leader-follower games

2024-11-01

Leader-follower games, also known as Stackelberg games, are models in game theory where one player (the leader) makes a decision first, and the other player (the follower) responds, considering the leader’s action. This is one the first instance of bilevel optimization.

Matrix mortality problem

2024-10-31

The Matrix Mortality Problem asks if a given set of square matrices can multiply to the zero matrix after a finite sequence of multiplications of elements. It is is undecidable for matrices of size 3x3 or larger. [ref]

Boolean satisfiability (SAT)

2024-10-30

The SAT problem asks whether a logical formula, composed of variables and (AND, OR, NOT), can be satisfied by assigning True to the variables. SAT is NP-complete along with 3-SAT, with clauses of three literals, while 2-SAT, is in P! [ref]

Krein—Milman theorem

2024-10-29

The Krein-Milman theorem states that any compact convex subset of a locally convex topological vector space is the closed convex hull of its extreme points. In particular, the set of extreme points of a nonempty compact convex set is nonempty. [ref]

Stone-Weierstrass theorem

2024-10-28

The Stone-Weierstrass theorem states that any continuous function on a compact Hausdorff space can be uniformly approximated by elements of a subalgebra, provided the subalgebra separates points and contains constants. [ref]

Nesterov Accelerated Gradient Descent

2024-10-25

The Nesterov Accelerated Gradient (NAG) algorithm refines gradient descent by using an extrapolation step before computing the gradient. It leads to faster convergence for smooth convex functions, achieving the optimal rate of O(1/k^2). [ref]

Pinhole camera model

2024-10-24

The pinhole camera model illustrates the concept of projective projection. Light rays from a 3D scene pass through an (infinitely) small aperture — the pinhole — and project onto a 2D surface, creating an inverted image on the focal plan. [ref]

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Łojasiewicz inequality

2024-10-23

Łojasiewicz inequality provides a way to control how close points are to the zeros of a real analytic function based on the value of the function itself. Extension of this result to semialgebraic or o-minimal functions exist. [ref]

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0x5F3759DF

2024-10-22

The fast inverse square root trick from Quake III is an algorithm to quickly approximate 1/√x, crucial for 3D graphics (normalisation). It uses bit-level manipulation and Newton's method for refinement. [ref]

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Monge problem: existence? uniqueness?

2024-10-21

A Monge map, i.e., a solution to optimal transport Monge problems, may not always exist, be unique, or be symmetric with respect to the source and target distributions. It was one of the motivation to introduce Kantorovich relaxation. [ref]

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Attouch's theorem

2024-10-18

Attouch’ theorem relates the (epi-) convergence of convex lsc functions to the convergence of their subdifferential seen as graph on the product between the space and its dual. [ref]

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Brenier's theorem

2024-10-17

Brenier's theorem states that the optimal transport map between two probability measures for quadratic cost is the gradient of a convex function. Moreover, it is uniquely defined up to a Lebesgue negligible set. [ref]

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Weierstrass function

2024-10-16

The Weierstrass function is a famous example of a continuous function that is nowhere differentiable. It defies intuition by being continuous everywhere but having no tangent at any point. Introduced by Karl Weierstrass in 1872.

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Proximal operator

2024-10-15

The proximal operator generalizes projection in convex optimization. It converts minimisers to fixed points. It is at the core of nonsmooth splitting methods and was first introduced by Jean-Jacques Moreau in 1965. [ref]

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Berry—Esseen Theorem (i.i.d version)

2024-10-14

The Berry-Esseen theorem quantifies how fast the distribution of the sum of independent random variables converges to a normal distribution, as described by the Central Limit Theorem (CLT). It provides an upper bound depending on skewness and sample size. [ref]

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Bilevel Optimization: optimistic vs pessimistic

2024-10-11

Bilevel optimization problems with multiple inner solutions come typically in two flavors: optimistic and pessimistic. Optimistic assumes the inner problem selects the best solution for the outer objective, while pessimistic assumes the worst-case solution is chosen.

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Geometry — according to Klein

2024-10-10

Klein geometry, part of the Erlangen Program introduced by F. Klein, studies geometry via transformation groups. A Klein geometry is defined as a space X=G/H, where G is a Lie group acting transitively on X. This unifies classical geometries under symmetry.[ref]

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Fenchel conjugate in 1D

2024-10-09

The Fenchel conjugate f*(y) is the maximum vertical gap between the line yx and the graph of f. This maximum occurs where the line is tangent to the curve, meaning f'(x) = y. It captures the largest gap at that tangent point.

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Gradient conjecture of R. Thom

2024-10-08

The Thom gradient conjecture states that the trajectory of a gradient flow of a real-analytic function must converge such that the secant converge also. It was proved by Kurdyka, Mostowski, and Parusiński in 2000. [ref]

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Johnson–Lindenstrauss lemma (1984)

2024-10-07

The Johnson–Lindenstrauss Lemma states that a set of high-dimensional points can be mapped into a much lower-dimensional space while approximately preserving pairwise distances. This is useful for dimensionality reduction, clustering, etc. [ref]

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Stein's lemma

2024-10-04

Stein's Lemma states that for a normally distributed variable X, the expected value E[Xg(X)] = E[g’(X)] for any g absolutely continuous (derivative a.e.) such that E[|g’(X)|] < ∞. It is a central result for characterizing (close-to) Gaussian data [ref]

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Erdős–Rényi(–Gilbert) graph models

2024-10-03

Erdős–Rényi-Gilbert graph models are two random models introduced in 1959 sharing a lot of common properties. In particular, G(n, p) exhibits a sharp threshold for the connectedness for p ≈ n/ln n. [ref]

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Law(s) of Large Numbers

2024-10-02

The law of large numbers tells that the empirical mean of an integrable random sample converges towards its mean in probability (weak LLN) and almost surely (strong LLN). [ref]

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Tarski—Seidenberg: logical statement

2024-10-01

The Tarski-Seidenberg theorem in logical form states that the set of first-order formulas over the real numbers is closed under quantifier elimination. This means any formula with quantifiers can be converted into an equivalent quantifier-free formula. [ref]

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Davis—Kahan sin(θ) inequality

2024-09-30

The Davis-Kahan sin(θ) theorem bounds the difference between the subspaces spanned by eigenvectors of two symmetric matrices, based on the difference between the matrices. It quantifies how small perturbations in a matrix affect its eigenvectors. [ref]

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Grid search

2024-09-27

Grid search is the most popular method for hyperparameter optimization in machine learning. Using a performance metric, it simply aims to “exhaustively” evaluate this metric on a discretisation of the sample space. It suffers from the curse of dimensionality.

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Determinant vs Permanent

2024-09-26

The determinant and permanent of a matrix may look similar, but they differ fundamentally: the determinant includes the signatures of permutations, while the permanent uses only addition. One is easy to compute, the others expected to be very hard. [ref]

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Chordal slope lemma

2024-09-25

If you pick three points along the curve in increasing order, the slope of the line between the first two points will always be less than or equal to the slope of the line between the last two points (chordal slope's lemma)

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Finite-difference in 1D (forward)

2024-09-24

Finite difference approximation is a numerical method to compute derivatives by discretization. The forward difference illustrated here is a first order approximation of the order of the stepsize. [ref]

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Barycenters in the Wasserstein space

2024-09-23

Barycenters in the Wasserstein space were studied by Agueh & Carlier in 2011. They define meaningful geometric mean for probability distribution, and are used in practice for instance for histogram transfer (Rabin et al. ’15) [ref]

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Tarski—Seidenberg: geometric statement

2024-09-20

Tarski—Seidenberg theorem claims that semialgebraic sets on 𝐑 are stable by projection. [ref]

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Semialgebraic set

2024-09-19

Semialgebraic sets can be expressed as finite union of polynomial equalities and inequalities. [ref]

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Banach fixed point & Picard’s iterations

2024-09-18

Banach fixed point theorem guarantees existence and uniqueness of fixed point of contraction mapping in metric spaces. It is at the root of the fixed point method.

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Dual and polar cones

2024-09-17

Dual cone and polar cone are sets in the dual of a vector space. They generalize the notion of orthogonal subspace.

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Voronoi diagram

2024-09-16

Voronoi diagram partitions the space in a proximity graph (cells) according to some sites. Each cell is the portion of the space where every points are closer to a given site than any other sites. [ref]

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Hamming distance

2024-09-13

Hamming distance is fundamental in coding theory: it corresponds to the number of different characters between two words of the same length. [ref]

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The rendering equation

2024-09-12

The rendering equation is the fundamental recursive integral equation of light transport proposed by Kajiya in 1986, at the heart of (ray) path-tracing and particle tracing. Numerous sampling methods have been proposed to solve it numerically. [ref]

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Random graph models

2024-09-11

Latent position models encompass popular random models of graphs such as SBM, Erdos-Renyi, etc. They are based on a latent space and a connectivity kernel, and strongly linked to graphons in the dense case. [ref]

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Clarke Generalized derivative

2024-09-10

Clarke derivative generalizes the notion of convex subdifferential to nonconvex locally Lipschitz functions — thanks to Rademacher’s theorem — as the convex hull of the limiting sequences of gradient converging. [ref]

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Complex step approximation

2024-09-09

Complex step approximation is a numerical method to approximate the derivative from a single function evaluation using complex arithmetic. It is some kind of “poor man” automatic differentiation. [ref]

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Stochastic Block Model

2024-09-06

Stochastic block model is a popular generative model for graph proposed by Holland-Laskey-Leinhardt in 1983 to model communities. [ref]

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Thomae's popcorn & Dirichlet functions

2024-09-05

Thomae’s popcorn function is a (1-periodic) discontinuous function on ℚ but continuous on ℝ\ℚ that is nowhere differentiable. Dirichlet is nowhere continuous on ℝ. [ref]

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Weisfeiler-Leman test

2024-09-04

(1-dimensional) Weisfeiler-Leman is a heuristic for the graph isomorphism problem. It is a color refinement algorithm akin to a basic message passing scheme. [ref]

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Talagrand's inequality

2024-09-03

Talagrand's inequality is a probabilistic isoperimetric inequality that allows to derive a concentration inequality for the median. This is an instance of "concentration of measure" that made him win the Abel Prize in 2024. [ref]

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Representation of closed set

2024-09-02

Every closed subset of 𝐑ⁿ is the zero set of a infinitely differentiable function from 𝐑ⁿ to 𝐑 (due to Whitney)

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