6.3 Applications in ML, Statistics, and Kernel Methods

6.3 Applications in ML, Statistics, and Kernel Methods

By now, we understand what Cholesky decomposition is and why it is efficient. But the deeper question—the one that determines whether this material matters to you as a practitioner—is much simpler:

Where does Cholesky actually show up in real systems?

The answer is: everywhere. Any time you see a covariance matrix, a kernel matrix, a quadratic form, or a Gaussian-based model, you are looking at something that depends on Cholesky—even if the library never exposes it directly. In machine learning, Bayesian inference, numerical optimization, and signal processing, Cholesky is the quiet engine making the system stable, fast, and reliable.

Gaussian Distributions: The Core of Statistics and ML

Gaussian models are the backbone of classical statistics and a surprising amount of machine learning. They appear in:

• Bayesian inference
• Kalman filters and smoothers
• Gaussian processes
• Variational inference
• PCA and factor analysis

The reason is simple:

A multivariate Gaussian requires Cholesky for everything: sampling, likelihood evaluation, parameter estimation, marginalization, and conditioning.

Given a covariance matrix Σ, we often need to compute:

• Σ-1 (never directly—always via solves)
• log det(Σ)
• samples x = μ + Lz with L the Cholesky factor
• Mahalanobis distances (x - μ)T Σ-1 (x - μ)

All of these use the Cholesky factorization. In fact, many statistical packages (Stan, PyMC, TFP, NumPyro, scikit-learn) rely so heavily on Cholesky that if it fails due to a non–positive-definite covariance matrix, the entire model collapses.

This is why modelers care so much about numerical stability in covariance matrices—because Cholesky is the gateway to the entire Gaussian world.

Gaussian Processes (GPs): Kernel Regression at Scale

Gaussian processes are one of the most elegant and powerful regression frameworks ever created, but they have one major bottleneck:

GP inference requires a Cholesky decomposition of the n × n kernel matrix.

A GP’s core operations are:

• K + σ2I → Cholesky factorization
• solve triangular systems for prediction
• compute log det through the Cholesky diagonal

The Cholesky factor is used for:

• training
• prediction
• hyperparameter optimization
• sampling from the posterior

And because the kernel matrix is symmetric positive definite (at least theoretically), Cholesky is the ideal tool—fast, stable, and memory efficient.

Even approximate GP techniques (like SKI, inducing points, random Fourier features) still rely on Cholesky decomposition for the smaller matrices involved.

Kernel Methods: The Hidden Structure of Many ML Algorithms

Kernel methods underpin:

• Support vector machines (SVMs)
• Kernel PCA
• Kernel ridge regression
• Spectral clustering
• Manifold learning

All these methods require the inversion (or solving) of a kernel matrix:

Solve (K + λI)α = y, where K is SPD.

The direct inversion is never computed. Instead, libraries perform:

LLT = K + λI

Then solve the triangular systems:

• Ly = b
• LTx = y

This process gives:

• maximum stability
• minimum memory usage
• fast log det evaluations
• efficient cross-validation

Without Cholesky, kernel methods would be too slow and too unstable to use in practice.

Kalman Filters and Smoothers

Kalman filters power:

• GPS tracking
• autonomous driving
• sensor fusion
• motion capture
• economic state models
• robotics

All forms of Kalman filtering (classic, extended, unscented, ensemble) rely on covariance updates that require:

• square roots of covariance matrices
• efficient solves involving the covariance

Many implementations use square-root filtering, which explicitly maintains a Cholesky factor of the covariance instead of the covariance itself. This improves:

• numerical stability
• performance
• filter reliability

When tracking systems fail or oscillate, it’s often because the covariance matrix became unstable and the Cholesky factorization failed.

Bayesian Modeling and MCMC

In Bayesian inference, Cholesky is everywhere:

• Hamiltonian Monte Carlo (HMC)
• No-U-Turn Sampler (NUTS)
• Laplace approximations
• Gaussian priors
• Random effects models
• Hierarchical models

Probabilistic programming frameworks (Stan, PyMC, NumPyro) call Cholesky thousands of times during sampling and optimization. It is the workhorse that:

• stabilizes transformations
• ensures valid covariance matrices
• supports reparameterized sampling

If you dig through their source code, you'll find that Cholesky is one of the most frequently optimized operations.

Quadratic Optimization and Machine Learning

Many optimization problems reduce to:

Solve Hx = g, where H is the Hessian or an approximation.

If H is SPD—as in least squares, Newton’s method, and many convex optimizers— then Cholesky is the best possible choice. It provides the fastest and most stable way to solve quadratic subproblems.

That includes:

• L-BFGS (via small SPD matrices)
• Levenberg–Marquardt
• Gauss–Newton
• Interior point methods
• Trust-region methods

Deep learning frameworks use Cholesky internally for the small but dense subproblems involved in advanced optimizers and second-order methods.

Covariance Estimation and Statistical Modeling

Estimating covariance matrices—especially in high dimensions—is notoriously difficult. Shrinkage estimators, graphical models, covariance regularization, and factor models all require robust ways to:

• invert covariance matrices
• compute determinants
• sample from distributions with complex covariance structure

These operations become unstable unless you use Cholesky decomposition. It is the only practical way to:

• compute log det reliably
• avoid pseudo-inversion errors
• test for positive-definiteness
• ensure stable sampling distributions

This is why nearly every statistical library includes Cholesky as a core primitive.

Why All These Applications Use Cholesky

A theme emerges across all these fields:

Whenever you see an SPD matrix, Cholesky is almost always the best tool—often the only tool.

It offers:

• maximal numerical stability
• minimal memory usage
• optimal performance
• clean mathematical structure
• simple integration with modern compute libraries

In practice, Cholesky ends up being used far more than LU or QR in many ML and statistical workflows. It is the backbone of efficient Gaussian modeling, kernel methods, optimization, and inference.

Where We Go Next

We’ve now explored LU, Cholesky, and the structure that enables efficient SPD computations. But not all matrices are symmetric or positive definite.

Some problems demand stability without symmetry. Others require solving least-squares problems. And some require orthogonal transformations to maintain numerical precision.

That brings us to the next major tool in numerical linear algebra:

QR Decomposition

QR is the algorithm you reach for when LU or Cholesky struggle—when you need stability above all else, when you want orthogonality to preserve numerical accuracy, or when solving least-squares problems at scale. Let’s dive into the world where orthogonal matrices become your best friend.