Markov Chain
A class of stochastic processes whose core property is that the conditional distribution of the current state depends only on a finite number of past states. A k-th order Markov chain depends on the most recent k states; a first-order chain satisfies the memoryless property that, given the current state, the future is independent of the past.
Core Facts
Timeline (last 90 days)
Previously the community generally believed that local algorithms (such as Markov chains) perform far worse than global algorithms in tensor PCA, and it was only known that under random initialization they require a polynomially higher signal-to-noise ratio to succeed.
A Markov chain describes a sequence of states in which each state depends only on the previous state and not on earlier history; this property is called the Markov property.
The reverse process of a diffusion model can only be approximated as a Markov chain when the number of noise-adding steps is sufficiently large and the noise at each step is sufficiently small.
The forward process of a diffusion model is a Markov chain, where each state depends only on the previous state.
Forward diffusion is a fixed Markov chain that gradually adds Gaussian noise to the data, and after T steps the data becomes approximately pure noise.
In KronLSTM, the evolution of the memory matrix can be described as a Markov chain and converges to a steady-state distribution.
This paper extends the classical least squares framework of source conditions and capacity conditions to the setting of data-dependent (Markov chain trajectory) sampling.
For a reversible Markov chain, the largest eigenvalue of the transition operator is 1, the absolute value of the second largest eigenvalue is denoted λ₂, and the spectral gap is defined as 1−|λ₂|.
A research paper analyzes the convergence of Markov chains under the framework of Hilbert space embedding geometry induced by positive definite kernels, namely Maximum Mean Discrepancy (MMD).
A k-th order Markov chain is a stochastic process in which the conditional distribution of the current state depends only on the most recent k historical states and is independent of earlier history.
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All Facts (12)
Forward diffusion is a fixed Markov chain that gradually adds Gaussian noise to the data, and after T steps the data becomes approximately pure noise.
70%UnverifiedPreviously the community generally believed that local algorithms (such as Markov chains) perform far worse than global algorithms in tensor PCA, and it was only known that under random initialization they require a polynomially higher signal-to-noise ratio to succeed.
50%UnverifiedA Markov chain describes a sequence of states in which each state depends only on the previous state and not on earlier history; this property is called the Markov property.
50%UnverifiedThe reverse process of a diffusion model can only be approximated as a Markov chain when the number of noise-adding steps is sufficiently large and the noise at each step is sufficiently small.
50%UnverifiedThe forward process of a diffusion model is a Markov chain, where each state depends only on the previous state.
50%UnverifiedIn KronLSTM, the evolution of the memory matrix can be described as a Markov chain and converges to a steady-state distribution.
50%UnverifiedThis paper extends the classical least squares framework of source conditions and capacity conditions to the setting of data-dependent (Markov chain trajectory) sampling.
50%UnverifiedFor a reversible Markov chain, the largest eigenvalue of the transition operator is 1, the absolute value of the second largest eigenvalue is denoted λ₂, and the spectral gap is defined as 1−|λ₂|.
50%UnverifiedA research paper analyzes the convergence of Markov chains under the framework of Hilbert space embedding geometry induced by positive definite kernels, namely Maximum Mean Discrepancy (MMD).
50%UnverifiedA k-th order Markov chain is a stochastic process in which the conditional distribution of the current state depends only on the most recent k historical states and is independent of earlier history.
50%UnverifiedThe paper proves that for finite-order Markov chains, the minimum memory length is upper-bounded by the Markov order.
50%UnverifiedMarkov chains and their variants remain continuously active as foundational tools for probabilistic inference, and have regained attention in the theoretical analysis of diffusion models.
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