We provide new upper bounds for mixing times of general finite Markov chains. We use these bounds to show that the total variation mixing time is robust under rough isometry for bounded degree graphs that are roughly isometric to trees.
Another variant on the Lov 'asz and Kannan result was obtained by Morris and Peres @cite_12 , who sharpened in several ways, including allowing non-reversible chains and bounding the larger @math -mixing time. This mantle was then taken up by Goel, Montenegro and Tet ali @cite_0 , who used the spectral profile @math instead of the conductance profile @math . For reversible chains @math may be thought of as the smallest eigenvalue of the Laplacian on the graph restricted to @math , minimized over all sets @math of invariant measure at most @math ; in fact the definition of @math used in @cite_0 is different, but for reversible chains it is equivalent to this definition up to constants provided that @math . Goel, Montenegro and Tet ali gave an upper bound involving @math , and were able to recover the result of Morris and Peres @cite_12 in the reversible case by using a discrete Cheeger inequality to relate @math to @math .
@cite_0 @cite_12
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Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset S such that its expansion (a.k.a. conductance) is bounded as follows: [ φ(S) def= (w(S,bar S )) (min set(w(S), w(bar(S)))) ≤ √(2 λ2) ] where w is the total edge weight of a subset or a cut and λ2 is the second smallest eigenvalue of the normalized Laplacian of the graph. Here we prove the following natural generalization: for any integer k ∈ [n], there exist ck disjoint subsets S1, ..., Sck, such that [ maxi φ(Si) ≤ C √(λk log k) ] where λk is the kth smallest eigenvalue of the normalized Laplacian and c 0 are suitable absolute constants. Our proof is via a polynomial-time algorithm to find such subsets, consisting of a spectral projection and a randomized rounding. As a consequence, we get the same upper bound for the small set expansion problem, namely for any k, there is a subset S whose weight is at most a O(1 k) fraction of the total weight and φ(S) ≤ C √(λk log k). Both results are the best possible up to constant factors. The underlying algorithmic problem, namely finding k subsets such that the maximum expansion is minimized, besides extending sparse cuts to more than one subset, appears to be a natural clustering problem in its own right. Consider a sequence of continuous-time irreducible reversible Markov chains and a sequence of initial distributions, @math . The sequence is said to exhibit @math -cutoff if the convergence to stationarity in total variation distance is abrupt, w.r.t. this sequence of initial distributions. In this work we give a characterization of @math -cutoff for an arbitrary sequence of initial distributions @math (in the above setup). Our characterization is expressed in terms of hitting times of sets which are "worst" w.r.t. @math . Consider a Markov chain on @math whose stationary distribution in @math . Let @math be the expected hitting time of the worst set of size at least @math . It was recently proved by Peres and Sousi and independently by Oliveira that @math captures the order of the mixing time. In this work we further refine this connection and show that @math -cutoff can be characterized in terms of concentration of hitting times (starting from @math ) of sets which are worst in expectation w.r.t. @math . Conversely, we construct a counter-example which demonstrates that in general cutoff (as opposed to cutoff w.r.t. a certain sequence of initial distributions) cannot be characterized in this manner. Finally, we also prove that there exists an absolute constant @math such that for every Markov chain @math , for all @math , where @math is the inverse of the spectral gap of the chain. The smallest eigenvalues and the associated eigenvectors (i.e., eigenpairs) of a graph Laplacian matrix have been widely used for spectral clustering and community detection. However, in real-life applications the number of clusters or communities (say, @math ) is generally unknown a-priori. Consequently, the majority of the existing methods either choose @math heuristically or they repeat the clustering method with different choices of @math and accept the best clustering result. The first option, more often, yields suboptimal result, while the second option is computationally expensive. In this work, we propose an incremental method for constructing the eigenspectrum of the graph Laplacian matrix. This method leverages the eigenstructure of graph Laplacian matrix to obtain the @math -th eigenpairs of the Laplacian matrix given a collection of all the @math smallest eigenpairs. Our proposed method adapts the Laplacian matrix such that the batch eigenvalue decomposition problem transforms into an efficient sequential leading eigenpair computation problem. As a practical application, we consider user-guided spectral clustering. Specifically, we demonstrate that users can utilize the proposed incremental method for effective eigenpair computation and determining the desired number of clusters based on multiple clustering metrics. This paper proves that an "old dog", namely -- the classical Johnson-Lindenstrauss transform, "performs new tricks" -- it gives a novel way of preserving differential privacy. We show that if we take two databases, @math and @math , such that (i) @math is a rank-1 matrix of bounded norm and (ii) all singular values of @math and @math are sufficiently large, then multiplying either @math or @math with a vector of iid normal Gaussians yields two statistically close distributions in the sense of differential privacy. Furthermore, a small, deterministic and alteration of the input is enough to assert that all singular values of @math are large. We apply the Johnson-Lindenstrauss transform to the task of approximating cut-queries: the number of edges crossing a @math -cut in a graph. We show that the JL transform allows us to that preserves edge differential privacy (where two graphs are neighbors if they differ on a single edge) while adding only @math random noise to any given query (w.h.p). Comparing the additive noise of our algorithm to existing algorithms for answering cut-queries in a differentially private manner, we outperform all others on small cuts ( @math ). We also apply our technique to the task of estimating the variance of a given matrix in any given direction. The JL transform allows us to that preserves differential privacy w.r.t bounded changes (each row in the matrix can change by at most a norm-1 vector) while adding random noise of magnitude independent of the size of the matrix (w.h.p). In contrast, existing algorithms introduce an error which depends on the matrix dimensions.