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One of the simplest algorithms for sampling from a probability distribution is Random Walk Metropolis-Hastings. It proposes new samples by taking Gaussian-distributed steps, accepting or rejecting them to maintain the target distribution. I call this pdf the "fidget spinner".

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Alec Helbling

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79,869 görüntüleme • 1 yıl önce •via X (Twitter)

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Flow Matching aims to learn a "flow" that transforms a simple source distribution (e.g. Gaussian) to an arbitrarily complex target distribution. This video shows the evolution of the marginal probability path as a source distribution is transformed to a target distribution.
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Flow Matching aims to learn a "flow" that transforms a simple source distribution (e.g. Gaussian) to an arbitrarily complex target distribution. This video shows the evolution of the marginal probability path as a source distribution is transformed to a target distribution.

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A basic strategy for drawing from a random distribution is to first generate uniformly-distributed points, then "warp" these points so they're spread out according to the target distribution. For example, the Box-Muller transform takes uniform points to normally-distributed ones
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A basic strategy for drawing from a random distribution is to first generate uniformly-distributed points, then "warp" these points so they're spread out according to the target distribution. For example, the Box-Muller transform takes uniform points to normally-distributed ones

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One common trick in MCMC methods is called 'burn-in'. It allows you to produce better samples by letting the chain first converge to the target distribution, discarding the first few samples which may be out of distribution.
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One common trick in MCMC methods is called 'burn-in'. It allows you to produce better samples by letting the chain first converge to the target distribution, discarding the first few samples which may be out of distribution.

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Langevin Monte Carlo allows you to draw samples from a probability distribution using its log gradient ∇ log p(x). By performing a sort of gradient ascent with noise you can navigate around the distribution. Langevin MC is heavily related to modern diffusion models.

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A discrete Markov chain is basically a random walk on a graph, where each outgoing edge has a fixed probability. Cool fact: any initial distribution on a sufficiently nice* Markov chain converges to its stationary distribution after many steps—because it's a contraction mapping.
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Hamiltonian Monte Carlo frames sampling from a probability distribution as a physics problem. By endowing "particles" with momentum and simulating their energy and motion through Hamilton's equations you can efficiently explore a distribution.

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This animation shows how a uniform grid of points is deformed by a flow matching model. The visualization highlights how the learned flow warps the underlying space—stretching, compressing, and bending it—to transform one probability distribution into another.
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Random Walks aim to explore a distribution through random perturbations. This is likely to move into low density regions, which is inefficient. Hamiltonian MC sails through a distribution much more rapidly by incorporating the distribution's structure into proposals.
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Random Walks aim to explore a distribution through random perturbations. This is likely to move into low density regions, which is inefficient. Hamiltonian MC sails through a distribution much more rapidly by incorporating the distribution's structure into proposals.

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This is a fun interactive article explaining the intuition behind Gaussian Processes. Gaussian Processes allow you to construct a distribution over functions with a continuous domain. They are incredibly useful for applications like Bayesian regression.
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I tried to capture this fidget spinner using the Huawei Pura70 Ultra to test its capability in handling fast-moving subjects. The result is truly impressive.
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The new Shader of the Week is "Fidget Cube" by TheBen27:

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Israel is wiping out the bridges of South Lebanon — one by one. Not for military necessity, but to isolate an entire population. To trap them. To starve them. To cut them off from hospitals, food, and aid. Call it what it is: A crime against humanity.
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