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Register a new userEstimating the Spectral Gap of a Reversible Markov Chain from a Short Trajectory
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The spectral gap $gamma$ of an ergodic and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $t$ may be observed. Hsu, Kontorovich, and Szepesvari (2015) considered the problem of estimating $gamma$ from this data. Let $pi$ be the stationary distribution of $P$, and $pi_star = min_x pi(x)$. They showed that, if $t = tilde{O}bigl(frac{1}{gamma^3 pi_star}bigr)$, then $gamma$ can be estimated to within multiplicative constants with high probability. They also proved that $tilde{Omega}bigl(frac{n}{gamma}bigr)$ steps are required for precise estimation of $gamma$. We show that $tilde{O}bigl(frac{1}{gamma pi_star}bigr)$ steps of the chain suffice to estimate $gamma$ up to multiplicative constants with high probability. When $pi$ is uniform, this matches (up to logarithmic corrections) the lower bound of Hsu, Kontorovich, and Szepesvari.
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We consider the three-state toric homogeneous Markov chain model (THMC) without loops and initial parameters. At time $T$, the size of the design matrix is $6 times 3cdot 2^{T-1}$ and the convex hull of its columns is the model polytope. We study the behavior of this polytope for $Tgeq 3$ and we show that it is defined by 24 facets for all $Tge 5$. Moreover, we give a complete description of these facets. From this, we deduce that the toric ideal associated with the design matrix is generated by binomials of degree at most 6. Our proof is based on a result due to Sturmfels, who gave a bound on the degree of the generators of a toric ideal, provided the normality of the corresponding toric variety. In our setting, we established the normality of the toric variety associated to the THMC model by studying the geometric properties of the model polytope.
Various problems in manifold estimation make use of a quantity called the reach, denoted by $tau_M$, which is a measure of the regularity of the manifold. This paper is the first investigation into the problem of how to estimate the reach. First, we study the geometry of the reach through an approximation perspective. We derive new geometric results on the reach for submanifolds without boundary. An estimator $hat{tau}$ of $tau_{M}$ is proposed in a framework where tangent spaces are known, and bounds assessing its efficiency are derived. In the case of i.i.d. random point cloud $mathbb{X}_{n}$, $hat{tau}(mathbb{X}_{n})$ is showed to achieve uniform expected loss bounds over a $mathcal{C}^3$-like model. Finally, we obtain upper and lower bounds on the minimax rate for estimating the reach.
We show that a large subclass of variograms is closed under products and that some desirable stability properties, such as the product of special compositions, can be obtained within the proposed setting. We introduce new classes of kernels of Schoenberg-L{e}vy type and demonstrate some important properties of rotationally invariant variograms.
The spectral gap $gamma$ of a finite, ergodic, and reversible Markov chain is an important parameter measuring the asymptotic rate of convergence. In applications, the transition matrix $P$ may be unknown, yet one sample of the chain up to a fixed time $n$ may be observed. We consider here the problem of estimating $gamma$ from this data. Let $pi$ be the stationary distribution of $P$, and $pi_star = min_x pi(x)$. We show that if $n = tilde{O}bigl(frac{1}{gamma pi_star}bigr)$, then $gamma$ can be estimated to within multiplicative constants with high probability. When $pi$ is uniform on $d$ states, this matches (up to logarithmic correction) a lower bound of $tilde{Omega}bigl(frac{d}{gamma}bigr)$ steps required for precise estimation of $gamma$. Moreover, we provide the first procedure for computing a fully data-dependent interval, from a single finite-length trajectory of the chain, that traps the mixing time $t_{text{mix}}$ of the chain at a prescribed confidence level. The interval does not require the knowledge of any parameters of the chain. This stands in contrast to previous approaches, which either only provide point estimates, or require a reset mechanism, or additional prior knowledge. The interval is constructed around the relaxation time $t_{text{relax}} = 1/gamma$, which is strongly related to the mixing time, and the width of the interval converges to zero roughly at a $1/sqrt{n}$ rate, where $n$ is the length of the sample path.
Markov chain Monte Carlo (MCMC) produces a correlated sample for estimating expectations with respect to a target distribution. A fundamental question is when should sampling stop so that we have good estimates of the desired quantities? The key to answering this question lies in assessing the Monte Carlo error through a multivariate Markov chain central limit theorem (CLT). The multivariate nature of this Monte Carlo error largely has been ignored in the MCMC literature. We present a multivariate framework for terminating simulation in MCMC. We define a multivariate effective sample size, estimating which requires strongly consistent estimators of the covariance matrix in the Markov chain CLT; a property we show for the multivariate batch means estimator. We then provide a lower bound on the number of minimum effective samples required for a desired level of precision. This lower bound depends on the problem only in the dimension of the expectation being estimated, and not on the underlying stochastic process. This result is obtained by drawing a connection between terminating simulation via effective sample size and terminating simulation using a relative standard deviation fixed-volume sequential stopping rule; which we demonstrate is an asymptotically valid procedure. The finite sample properties of the proposed method are demonstrated in a variety of examples.
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