This is a mirror of the daily Samuel Vaiter's serie available on bluesky. One piece of math = one slide = one post every (working) day. All content is available under the CC BY-SA license.
2025-03-10
Transformers revolutionized first NLP, then the whole ML world. It is built upon celebrated concepts like feedforward networks, normalization, residual connection, clever embeddings strategies, and the now essential (multi-head) attention mechanism. [ref]
2025-02-28
The Poisson process is a counting process that models the number of “rare” events occurring in a fixed interval of time or space. It has stationary and independent increments, and the time between events follows an exponential distribution.
2025-02-27
The Brunn–Minkowski theorem states that for nonempty compact set in Rn, the volume of their Minkowski sum is at least the sum of the nth roots of their volumes. It implies many inequalities in convex geometry including the isoperimetric inequality.
2025-02-26
ResNet was a major milestone in deep learning by (re-)introducing skip connections, where networks try to learn residual mappings instead of identity. This allows very deep models, and is used in most modern architectures, including Transformers. [ref]
2025-02-25
The Hausdorff distance quantifies how far two sets are from each other by measuring the largest minimal distance needed to cover one with the other. It captures worst-case deviation: if one set has points far from the other, the distance is large.
2025-02-24
Signed Distance Functions/Fields define shapes implicitly by measuring the shortest distance from any point to the surface. Negative values are inside, and positive values are outside. SDFs allow smooth blending, Boolean operations for procedural modeling. [ref]
2025-02-07
The McKean-Vlasov equation drives stochastic processes where the drift and diffusion depend on both the state and the law of the process. It can be seen as the limit of interacting particle system, connecting stochastic dynamics with nonlinear Fokker-Planck equations. [ref]
2025-02-06
To get token embeddings, text is tokenized into tokens (e.g., via BPE). Each token is assigned a unique ID. IDs are converted to one-hot vectors, then multiplied by an embedding matrix (learned during training) to produce dense, fixed-size token embeddings.
2025-02-05
Mathematical morphology studies shapes and structures in images using tools like erosion, dilation, opening, and closing. It was introduced by Matheron & Serra in the 60s. It is typically defined on abstract lattices. [ref]
2025-02-04
The Rudin-Osher-Fatemi (ROF) model is a method in image denoising. It minimizes the total variation of an image (preserving edges) while trying to be close in the L2 sense to the observations. [ref]
2025-02-03
Natural Language Processing is a broad field with several applications: next-token prediction, fill-in-the-middle, sentiment analysis, question answering, speech recognition (and text-to-speech), translation, etc. [ref]
2025-01-31
Breadth-First Search (BFS) explores all nodes level by level, using a queue. Depth-First Search (DFS) goes deep into one branch before backtracking, using a stack/recursion.
2025-01-30
Byte Pair Encoding is a tokenization method that starts with all characters as initial tokens. It iteratively merges the most frequent adjacent byte pairs in the text, adding new tokens to the vocabulary until reaching a predefined size. The output is a sequence of subword tokens. [ref]
2025-01-29
Cyclic projection onto convex sets is a method to find a point in the intersection of convex sets. It iteratively projects onto each set in a cycle, and it converges to a point in the intersection under restrictive assumptions. [ref]
2025-01-28
A single-head attention layer processes a sequence by assigning relevance scores between elements. It computes these scores to decide how much focus each part of the seq should receive and combines the information. [ref]
2025-01-27
Zeros of random polynomials with i.i.d. coefficients exhibit interesting convergence properties. In particular, normal coefficient leads to weak convergence of the empirical measure to the unit circle. [ref]
2025-01-24
The Neural Probabilistic Language Model (Bengio et al., 2000) introduced joint use of word embeddings and neural networks for next-token prediction. It replaced n-gram models by mapping words to a continuous vector space, and the use a standard neural networks architectures. [ref]
2025-01-23
ResNet and Neural ODEs are closely related: ResNet uses discrete residual/skip connections, while Neural ODEs generalize this to continuous transformations using ODEs. Neural ODEs can be seen as the limit of ResNet as the number of layers approaches infinity. [ref]
2025-01-22
The Marchenko–Pastur law describes the limiting spectral distribution of eigenvalues of large random covariance matrices. The theorem proves that as dimensions grow, the empirical spectral distribution converges weakly to this law. [ref]
2025-01-21
Singular Value Decomposition is a matrix factorization that expresses a matrix M as UΣVᵀ, where U and V are orthogonal matrices, and Σ is diagonal. It can interpreted as the action of a rotation, a scaling and a rotation/reflection. [ref]
2025-01-20
Napoleon’s problem asks to construct the center of a circle with only a compass. It is possible to solve it with six circles. [ref]
2025-01-17
Soft(arg)max function transforms logits into probability vector. One shows that softmax (w/ temperature) converges pointwise to the argmax, selecting the most likely outcome. Essential in in multinomial logistic regression and for attention mechanisms
2025-01-16
The Cheap Gradient Principle (Baur—Strassen, 1983) states that computing gradients via automatic differentiation is efficient: the gradient of a function f can be obtained with a cost proportional to evaluating f
2025-01-15
A group is one of the most fundamental structure in algebra: a set paired with a binary operation satisfying associativity, identity, and invertibility. Exemples include the real line, integers with addition, modular addition, symmetric group, rotations, etc.
2025-01-14
Topological order is a total (all elements are comparable) order of vertices in a directed acyclic graph where for every directed edge u→v, vertex u appears before v in the order. It’s used in tasks like scheduling, circuit design and automatic differentiation.
2025-01-13
Heath-Brown/Zagier proofs of Fermat’s sum of two squares theorem relies on the fact that the cardinalities of a set and the fixed points of an involution on it have the same parity. [ref]
2025-01-10
Algorithm (asymptotic) complexity typically measures time or space required as input size grows. They are measured against a model of computation, typically a Turing machine or variations.
2025-01-09
Tropical geometry studies algebraic varieties through the “tropical semiring” where addition is min (or max) and multiplication is the regular addition. It transforms polynomial eqs into piecewise linear structures and is linked to dynamic programming. [ref]
2025-01-08
Brownian motion is the fundamental martingale that arises in almost all areas of mathematics with a stochastic aspect, used in biology, passing by statistical mechanics and diffusion model. [ref]
2025-01-07
The Nadaraya-Watson estimator is linear local averaging estimator relying on a pointwise nonnegative kernel. Most of the time, a box or Gaussian kernel is used. [ref]
2025-01-06
The concept of partly smooth functions (Lewis ‘02) unifies many nonsmooth functions, capturing their geometry: smooth along the manifold and sharp normal to it. It allows fine sensitivity analysis results. In the convex case, it generalizes sparsity.
2025-01-03
Complex transformation lead to fun visual effects. Inversion is a transformation that maps a point to its reflection through a circle. It is conformal, meaning it preserves angles. [ref]
2025-01-02
The Game of Life is a cellular automaton devised by John Conway in 1970. It consists of a grid of cells that evolve according to simple deterministic rules. Despite the simplicity of the rules, the Game of Life can exhibit complex behavior. [ref]
2025-01-01
Smoothstep functions are a family of functions that interpolate between 0 and 1 smoothly. They are used in computer graphics to create smooth transitions between colors or shapes. [ref]
2024-12-31
The Mandelbrot set is a fractal defined by iterating the complex map z² + c. Zooming in reveals patterns and self-similarity at all scales. [ref]
2024-12-30
The Sierpinski triangle is a fractal with the property of self-similarity. It is constructed by iteratively removing the central triangle of each triangle in a sequence of triangles. [ref]
2024-12-27
A cyclide is a surface that is a generalization of the torus and can be defined as the envelope of a family of spheres.
2024-12-26
The equation x² − y²z = 0 describes the Whitney umbrella. It is a ruled surface with a cusp at the origin.
2024-12-25
Fermat’s theorem on sums of two squares is a result of number theory that was first described in a letter of Fermat to Mersenne on December 25, 1640. It was first proved a century later by Euler in 1749. [ref]
2024-12-24
The Möbius strip is a one-sided surface with no boundaries, formed by twisting a strip of paper 180° and joining its ends.
2024-12-23
The algebraic surface (x+y+z)² + xyz = 0 is one of Cayley singular cubic. Cayley classifies them in 23 cases. [ref]
2024-12-20
A spirograph is a parametric curve generated by tracing a point fixed on a circle as it rolls along the inside or outside of another circle. The resulting pattern depends on the radius of the circles and the point’s position. [ref]
2024-12-19
When optimization problems have multiple minima, algorithms favor specific solutions due to their implicit bias. For ordinary least squares (OLS), gradient descent inherently converges to the minimal norm solution among all possible solutions. [ref]
2024-12-18
Linearly inducible orderings of n points in d-dimensional space are determined by projections onto a reference vector. Their combinatorial are typically less than the number of permutations n! (except when d ≥ n-1). [ref]
2024-12-17
There exists f:[0,1]→[0,1] strictly increasing, continuous function such that its derivative is 0 almost everywhere. [ref]
2024-12-16
Gauss-Lucas theorem states that the roots of the derivative of a polynomial with complex coefficients always lie within the convex hull of the original polynomial’s roots.
2024-12-13
Automatic differentiation in forward mode computes derivatives by breaking down functions into elem operations and propagating derivatives alongside values. It’s efficient for functions with fewer inputs than outputs and for Jacobian-vect prod, using for instance dual numbers.
2024-12-12
The Sinkhorn—Knopp algorithm is an iterative method for approximating entropic optimal transport solutions. It balances two dual vectors through iterative rescaling leading to a matrix that respects marginals. Thanks to its parallel nature, it’s easy to implement on GPU for ML.
2024-12-11
Perlin’s noise generates smooth textures by assigning random gradient vectors to grid points and blending values through smooth interpolation. It is used in graphics, terrain, and textures generation. [ref]
2024-12-10
Newton’s method is an iterative root-finding algorithm that uses the derivative of a function to approximate a root. It converges quadratically, but may fail to converge or converge to a wrong root if the initial guess is not close enough.
2024-12-09
Kuratowski’s theorem states that a graph is planar (can be drawn on a plane without edge crossings) iff it does not contain a subgraph that is a subdivision of either K₅ or K₃,₃. [ref]
2024-12-06
A zonoid is a convex body that is a limit of zonotope (polytopes formed by the Minkowski sum of line segments). Bourgain et al. gave a precise number of segments required to approximate a zonoid by a zonotope in Hausdorff metric for a precision. [ref]
2024-12-05
Morley’s theorem (1899) states that the intersections of the adjacent angle trisectors of any triangle defines an equilateral triangle. [ref]
2024-12-04
Classical trinom proof of Cauchy-Schwarz is boring! It can be proved using only the squared norm of the addition. [ref]
2024-12-03
There are a lot of different datasets in computer vision, with various dimensions, number of classes, text annotations or different kind of structural information (e.g., bounding boxes). Some popular general datasets are MNIST, CIFAR10/100, Pascal VOC, ImageNet and COCO.
2024-12-02
Birkhoff’s contraction theorem states that linear nonnegative (i.e. such that the image of the cone is contained in itself) mappings are contraction with respect to the Hilbert metric. In particular, it allows to use Banach fixed point theorem .[ref]
2024-11-29
A bigram model is a language model that predicts the next token based only on the previous one. It is an example of a discrete Markov chain, and was in fact one of the first motivation of Markov himself. [ref]
2024-11-28
When trying to compute a dual of a composite problem involving two functions and two linear operators (e.g., Total Variation regularization of inverse problems), it is sometimes useful to consider either of the operators as the dual operator.
2024-11-27
Shannon’s entropy measures the uncertainty or information content in a probability distribution. It’s an important concept in data compression and communication introduced in the seminal paper “A Mathematical Theory of Communication”. [ref]
2024-11-26
The Banach-Steinhaus theorem, or Uniform Boundedness Principle, is a key result in functional analysis. It states that for a family of continuous linear operators on a Banach space, pointwise boundedness implies uniform boundedness. [ref]
2024-11-25
The Hilbert projective (constant on rays) metric is a way to measure distances within a convex cone in a vector space. It is an important tool in hyperbolic geometry and for Perron-Frobenius theory. [ref]
2024-11-22
Mersenne primes are of the form Mₙ = 2ⁿ - 1, where n itself is a prime number. The Lucas-Lehmer test is an efficient method to check if Mₙ is prime. It iterates a sequence mod Mₙ to determine primality as used by the GIMPS project. [ref]
2024-11-21
Bubble Sort is an adaptative sorting algorithm that compares adjacent items, and swaps them if they’re in the wrong order. It is easy to understand and implement, but inefficient in both the worst and average cases. [ref]
2024-11-20
Tokenization is the process of breaking text into smaller units - tokens - which can be words, subwords, or characters. It’s a step in NLP, transforming raw text into token embeddings. The classical algorithm is Byte-Pair Encoding at the subword level. [ref]
2024-11-19
The Lorenz system is a model that describes chaotic behavior in atmospheric convection. Its study reveals how small changes in initial conditions can lead to different outcomes. [ref]
2024-11-18
Covering numbers can be thought as way to quantify “compactness” of a set in a complete metric space. It is also closely related to packing numbers. [ref]
2024-11-15
Hutchinson’s estimator is a stochastic method for approximating the trace of large square matrices. It requires only matrix-vector products, allowing to compute the trace of implicitly defined matrices. It is useful in several settings (cross-validation, trace of the Hessian, etc). [ref]
2024-11-14
Graph isomorphism problem asks the following question: given two finite graphs, are they isomorphic? There is no definite answer if it is in P, NP-complete or in another class (for the moment, it defines its proper complexity class GI). [ref] (in french)
2024-11-13
The cumulative distribution function (CDF) gives the probability that a random variable is less than or equal to a value. It characterizes a distribution. The quantile function (generalized inverse of the CDF) returns the value for a given cumulative probability.
2024-11-12
The Universal Approximation Theorem states that a feedforward neural network with a single hidden layer, using a non-linear activation function, can approximate any continuous function on a compact, given enough neurons. [ref]
2024-11-11
Prim’s algorithm is a greedy method for finding the minimum spanning tree of a weighted, connected graph. It starts with any node and repeatedly adds the smallest edge that connects a visited node to an unvisited node. [ref]
2024-11-08
Convergence of random variables (and associated probability measures) comes in several modes: almost sure, Lp, in probability, but also in Wasserstein distance or Total Variation in the space of measures. [ref]
2024-11-07
Harmonic functions solve Laplace’s equation Δu = 0 and are infinitely differentiable (and analytic). They exhibit no local maxima or minima within the domain, achieving extrema only on the boundary. Under Dirichlet boundary conditions, Laplace solution is uniquely determined. [ref]
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]
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]
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]
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.
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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]
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.
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]
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]
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.
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]
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.
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]
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]
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]
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]
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]
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]
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]
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.
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]
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)
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]
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]
2024-09-20
Tarski—Seidenberg theorem claims that semialgebraic sets on 𝐑 are stable by projection. [ref]
2024-09-19
Semialgebraic sets can be expressed as finite union of polynomial equalities and inequalities. [ref]
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.
2024-09-17
Dual cone and polar cone are sets in the dual of a vector space. They generalize the notion of orthogonal subspace.
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]
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]
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]
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]
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]
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]
2024-09-06
Stochastic block model is a popular generative model for graph proposed by Holland-Laskey-Leinhardt in 1983 to model communities. [ref]
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]
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]
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]
2024-09-02
Every closed subset of 𝐑ⁿ is the zero set of a infinitely differentiable function from 𝐑ⁿ to 𝐑 (due to Whitney)
