By viewing the @math -simplex as the set of positions of @math ordered particles on the unit interval, the adjacent walk is the continuous time Markov chain obtained by updating independently at rate 1 the position of each particle with a sample from the uniform distribution over the interval given by the two particles adjacent to it. We determine its spectral gap and prove that both the total variation distance and the separation distance to the uniform distribution exhibit a cutoff phenomenon, with mixing times that differ by a factor @math . The results are extended to the family of log-concave distributions obtained by replacing the uniform sampling by a symmetric log-concave Beta distribution.
Let us briefly describe the main arguments used to establish the results of this article. Concerning the spectrum, the first observation is that when restricted to the invariant subspace of linear functions, the generator can be easily diagonalised. In particular, one finds that @math is an eigenfunction of @math with eigenvalue @math and therefore, for all @math , the spectral gap is at most @math . Proving that this is actually the spectral gap is more delicate. We prove it only in the case @math since we rely on the FKG inequality. In this case, using the strict monotonicity of @math , we show that any eigenfunction associated to an eigenvalue larger than @math needs to be non-decreasing as well, and the FKG inequality is then applied to conclude that the only such eigenfunction is the constant one; see Section below. Concerning the case @math , we can only assert that the spectral gap is not smaller than @math for some constant @math . Indeed, this is a direct consequence of the mean field spectral gap and a comparison argument (see e.g. @cite_18 ).
@cite_18
2889975307 2114789426 2949688822 2589022458
Spectral algorithms, such as principal component analysis and spectral clustering, typically require careful data transformations to be effective: upon observing a matrix @math , one may look at the spectrum of @math for a properly chosen @math . The issue is that the spectrum of @math might be contaminated by non-informational top eigenvalues, e.g., due to scale variations in the data, and the application of @math aims to remove these. Designing a good functional @math (and establishing what good means) is often challenging and model dependent. This paper proposes a simple and generic construction for sparse graphs, @math where @math denotes the adjacency matrix and @math is an integer (less than the graph diameter). This produces a graph connecting vertices from the original graph that are within distance @math , and is referred to as graph powering. It is shown that graph powering regularizes the graph and decontaminates its spectrum in the following sense: (i) If the graph is drawn from the sparse Erd o s-R 'enyi ensemble, which has no spectral gap, it is shown that graph powering produces a maximal' spectral gap, with the latter justified by establishing an Alon-Boppana result for powered graphs; (ii) If the graph is drawn from the sparse SBM, graph powering is shown to achieve the fundamental limit for weak recovery (the KS threshold) similarly to massoulie-STOC , settling an open problem therein. Further, graph powering is shown to be significantly more robust to tangles and cliques than previous spectral algorithms based on self-avoiding or nonbacktracking walk counts massoulie-STOC,Mossel_SBM2,bordenave,colin3 . This is illustrated on a geometric block model that is dense in cliques. We consider the problem of robustly recovering a @math -sparse coefficient vector from the Fourier series that it generates, restricted to the interval @math . The difficulty of this problem is linked to the superresolution factor SRF, equal to the ratio of the Rayleigh length (inverse of @math ) by the spacing of the grid supporting the sparse vector. In the presence of additive deterministic noise of norm @math , we show upper and lower bounds on the minimax error rate that both scale like @math , providing a partial answer to a question posed by Donoho in 1992. The scaling arises from comparing the noise level to a restricted isometry constant at sparsity @math , or equivalently from comparing @math to the so-called @math -spark of the Fourier system. The proof involves new bounds on the singular values of restricted Fourier matrices, obtained in part from old techniques in complex analysis. We describe the spectral statistics of the first finite number of eigenvalues in a newly-forming band on the hard-edge of the spectrum of a random Hermitean matrix model. It is found that in a suitable scaling regime, they are described by the same spectral statistics of a finite-size Laguerre-type matrix model. The method is rigorously based on the Riemann-Hilbert analysis of the corresponding orthogonal polynomials. We consider the following detection problem: given a realization of a symmetric matrix X of dimension n, distinguish between the hypothesis that all upper triangular variables are i.i.d. Gaussians variables with mean 0 and variance 1 and the hypothesis that there is a planted principal submatrix B of dimension L for which all upper triangular variables are i.i.d. Gaussians with mean 1 and variance 1 , whereas all other upper triangular elements of X not in B are i.i.d. Gaussians variables with mean 0 and variance 1. We refer to this as the 'Gaussian hidden clique problem'. When L = (1 + ∊) √n (∊ > 0), it is possible to solve this detection problem with probability 1 - on(1) by computing the spectrum of X and considering the largest eigenvalue of X. We prove that when L < (1 - ∊) √n no algorithm that examines only the eigenvalues of X can detect the existence of a hidden Gaussian clique, with error probability vanishing as n → ∞. The result above is an immediate consequence of a more general result on rank-one perturbations of k-dimensional Gaussian tensors. In this context we establish a lower bound on the critical signal-to-noise ratio below which a rank-one signal cannot be detected.