We prove deviation bounds for the random variable @math in which @math is a Markov chain with stationary distribution and state space @math , and @math . Our bound improves upon previously known bounds in that the dependence is on @math rather than @math We also prove deviation bounds for certain types of sums of vector--valued random variables obtained from a Markov chain in a similar manner. One application includes bounding the expected value of the Schatten @math -norm of a random matrix whose entries are obtained from a Markov chain.
In work by Garg, Lee, Song and Srivastava @cite_13 , a version of Eq. was proved for Markov chains when the Banach space is the set of @math matrices under the Schatten @math -norm, generalizing a result first shown by Ahlswede and Winter @cite_17 (see also the monograph by Tropp @cite_9 ). Note that the Schatten @math -norm of a @math matrix is up to constant factors equal to the Schatten @math -norm of that matrix, and the modulus of uniform smoothness of the set of @math matrices under the Schatten @math -norm is @math . Thus in this case the left-hand side of Eq. is bounded above by @math . Garg, Lee, Song and Srivastava showed that for a Markov chain @math and functions @math such that @math ,
@cite_9 @cite_13 @cite_17
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Given an @math matrix @math , its Schatten- @math norm, @math , is defined as @math , where @math is the @math -th largest singular value of @math . These norms have been studied in functional analysis in the context of non-commutative @math -spaces, and recently in data stream and linear sketching models of computation. Basic questions on the relations between these norms, such as their embeddability, are still open. Specifically, given a set of matrices @math , suppose we want to construct a linear map @math such that @math for each @math , where @math and @math , and further, @math for a given approximation factor @math and real number @math . Then how large do @math and @math need to be as a function of @math ? We nearly resolve this question for every @math , for the case where @math can be expressed as @math , where @math and @math are arbitrary matrices that are allowed to depend on @math , that is, @math can be implemented by left and right matrix multiplication. Namely, for every @math , we provide nearly matching upper and lower bounds on the size of @math and @math as a function of @math . Importantly, our upper bounds are oblivious , meaning that @math and @math do not depend on the @math , while our lower bounds hold even if @math and @math depend on the @math . As an application of our upper bounds, we answer a recent open question of about space-approximation trade-offs for the Schatten @math -norm, showing in a data stream it is possible to estimate the Schatten- @math norm up to a factor of @math using @math space. We consider the problem of selecting the best subset of exactly @math columns from an @math matrix @math . We present and analyze a novel two-stage algorithm that runs in @math time and returns as output an @math matrix @math consisting of exactly @math columns of @math . In the first (randomized) stage, the algorithm randomly selects @math columns according to a judiciously-chosen probability distribution that depends on information in the top- @math right singular subspace of @math . In the second (deterministic) stage, the algorithm applies a deterministic column-selection procedure to select and return exactly @math columns from the set of columns selected in the first stage. Let @math be the @math matrix containing those @math columns, let @math denote the projection matrix onto the span of those columns, and let @math denote the best rank- @math approximation to the matrix @math . Then, we prove that, with probability at least 0.8, @math This Frobenius norm bound is only a factor of @math worse than the best previously existing existential result and is roughly @math better than the best previous algorithmic result for the Frobenius norm version of this Column Subset Selection Problem (CSSP). We also prove that, with probability at least 0.8, @math This spectral norm bound is not directly comparable to the best previously existing bounds for the spectral norm version of this CSSP. Our bound depends on @math , whereas previous results depend on @math ; if these two quantities are comparable, then our bound is asymptotically worse by a @math factor. Let (a>0, b>0, ab<1; ) and let (g L^2( R ). ) In this paper we investigate the relation between the frame operator (S:f L^2( R ) n,m ,(f,g_ na,mb ) ,g_ na,mb ) and the matrix (H ) whose entries (H_ k,l ,; ,k',l' ) are given by ((g_ k' b,l' a ,g_ k b,l a ) ) for (k,l,k',l' Z . ) Here (f_ x,y (t)= exp (2 iyt) ,f(t-x), ) (t R ), for any (f L^2( R ). ) We show that (S ) is bounded as a mapping of (L^2( R ) ) into (L^2( R ) ) if and only if (H ) is bounded as a mapping of (l^2( Z ^2) ) into (l^2( Z ^2). ) Also we show that (AI S BI ) if and only if (AI 1 ab ,H BI, ) where (I ) denotes the identity operator of (L^2( R ) ) and (l^2( Z ^2), ) respectively, and (A 0, ) (B< . ) Next, when (g ) generates a frame, we have that ((g_ k b,l a )_ k,l ) has an upper frame bound, and the minimal dual function (^ ) can be computed as (ab , k,l ,(H^ -1 )_ k,l ,; ,o,o ,g_ k b,l a . ) The results of this paper extend, generalize, and rigourize results of Wexler and Raz and of Qian, D. Chen, K. Chen, and Li on the computation of dual functions for finite, discrete-time Gabor expansions to the infinite, continuous-time case. Furthermore, we present a framework in which one can show that certain smoothness and decay properties of a (g ) generating a frame are inherited by (^ . ) In particular, we show that (^ S ) when (g S ) generates a frame (( S ) Schwartz space). The proofs of the main results of this paper rely heavily on a technique introduced by Tolimieri and Orr for relating frame bound questions on complementary lattices by means of the Poisson summation formula. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions.