https://juancopi81.github.io/writing.html Research logs, derivations, and tutorials from personal machine learning projects, including an ongoing series rebuilding diffusion models from first principles. en-us Thu, 01 Oct 2026 00:00:00 +0000 https://juancopi81.github.io/posts/2026-10-01-when-the-vae-meets-monte-carlo.html https://juancopi81.github.io/posts/2026-10-01-when-the-vae-meets-monte-carlo.html Thu, 01 Oct 2026 00:00:00 +0000 Why averaging sampled reconstruction losses estimates an expectation in a VAE, through the law of large numbers and an indicator-variable approach to integration. https://juancopi81.github.io/posts/2026-09-17-in-search-of-pcas-lost-variance.html https://juancopi81.github.io/posts/2026-09-17-in-search-of-pcas-lost-variance.html Thu, 17 Sep 2026 00:00:00 +0000 A geometric derivation of why PCA reconstruction mean-squared error equals the sum of the discarded eigenvalues, through orthogonal projections and a 2D example. https://juancopi81.github.io/posts/2026-09-03-forward-process-transformations-and-gregor-samsa.html https://juancopi81.github.io/posts/2026-09-03-forward-process-transformations-and-gregor-samsa.html Thu, 03 Sep 2026 00:00:00 +0000 A derivation and geometric playground connecting the affine Gaussian transformation in a diffusion forward process to covariance and the Jacobian determinant. https://juancopi81.github.io/posts/2026-08-13-sampling-our-way-down-from-gaussian-distributions.html https://juancopi81.github.io/posts/2026-08-13-sampling-our-way-down-from-gaussian-distributions.html Thu, 13 Aug 2026 00:00:00 +0000 A small-step Bayes derivation showing why the reverse conditional can be approximated by a Gaussian even when the previous-state marginal is not Gaussian. https://juancopi81.github.io/posts/2026-08-06-kl-divergence-is-not-a-distance.html https://juancopi81.github.io/posts/2026-08-06-kl-divergence-is-not-a-distance.html Thu, 06 Aug 2026 00:00:00 +0000 A daily-life example using car and train commute-time distributions to understand why KL divergence is asymmetric and how it can be used to compare distributions. https://juancopi81.github.io/posts/2026-07-30-two-ways-to-learn-the-same-score-field.html https://juancopi81.github.io/posts/2026-07-30-two-ways-to-learn-the-same-score-field.html Thu, 30 Jul 2026 00:00:00 +0000 A small experiment comparing two ways to train a toy neural network to learn the score of a noisy three-component Gaussian mixture. https://juancopi81.github.io/posts/2026-07-23-averaging-paths-to-the-peaks-of-our-mountains.html https://juancopi81.github.io/posts/2026-07-23-averaging-paths-to-the-peaks-of-our-mountains.html Thu, 23 Jul 2026 00:00:00 +0000 A derivation of how the noisy marginal score connects to the posterior conditional mean through an average over plausible clean samples. https://juancopi81.github.io/posts/2026-07-16-transforming-random-variables.html https://juancopi81.github.io/posts/2026-07-16-transforming-random-variables.html Thu, 16 Jul 2026 00:00:00 +0000 A short derivation of the density of a transformed random variable, with intuition for how the Jacobian compensates for stretched lengths and areas. https://juancopi81.github.io/posts/2026-07-09-when-the-score-does-not-point-straight-to-the-mean.html https://juancopi81.github.io/posts/2026-07-09-when-the-score-does-not-point-straight-to-the-mean.html Thu, 09 Jul 2026 00:00:00 +0000 An interactive look at how covariance reshapes the score of a multivariate Gaussian, and why the direction of higher density does not always point straight to the mean. https://juancopi81.github.io/posts/2026-07-02-denoising-is-not-inversion.html https://juancopi81.github.io/posts/2026-07-02-denoising-is-not-inversion.html Thu, 02 Jul 2026 00:00:00 +0000 A conditional-expectation note on why denoising under squared error is estimation, not exact inversion. https://juancopi81.github.io/posts/2026-06-25-score-of-a-gaussian.html https://juancopi81.github.io/posts/2026-06-25-score-of-a-gaussian.html Thu, 25 Jun 2026 00:00:00 +0000 A derivation-driven note on why the score of a one-dimensional Gaussian points toward higher log-density, which means back toward the mean. https://juancopi81.github.io/posts/2026-06-11-expected-value-simple-forward-process.html https://juancopi81.github.io/posts/2026-06-11-expected-value-simple-forward-process.html Thu, 11 Jun 2026 00:00:00 +0000 A diffusion note on what it really means for noisy images to stay centered around the original image on average. https://juancopi81.github.io/posts/2026-06-04-bernoulli-sequences-same-probability.html https://juancopi81.github.io/posts/2026-06-04-bernoulli-sequences-same-probability.html Thu, 04 Jun 2026 00:00:00 +0000 A first-principles probability note on why conditioning on the total count makes all Bernoulli sequences with exactly k successes equally likely.