30 research outputs found
On the sub-Gaussianity of the Beta and Dirichlet distributions
We obtain the optimal proxy variance for the sub-Gaussianity of Beta
distribution, thus proving upper bounds recently conjectured by Elder (2016).
We provide different proof techniques for the symmetrical (around its mean)
case and the non-symmetrical case. The technique in the latter case relies on
studying the ordinary differential equation satisfied by the Beta
moment-generating function known as the confluent hypergeometric function. As a
consequence, we derive the optimal proxy variance for the Dirichlet
distribution, which is apparently a novel result. We also provide a new proof
of the optimal proxy variance for the Bernoulli distribution, and discuss in
this context the proxy variance relation to log-Sobolev inequalities and
transport inequalities.Comment: 13 pages, 2 figure
On the Accuracy of Hotelling-Type Asymmetric Tensor Deflation: A Random Tensor Analysis
This work introduces an asymptotic study of Hotelling-type tensor deflation
in the presence of noise, in the regime of large tensor dimensions.
Specifically, we consider a low-rank asymmetric tensor model of the form
where and
the 's are unit-norm rank-one tensors such that for and is an additive noise term. Assuming that the dominant
components are successively estimated from the noisy observation and
subsequently subtracted, we leverage recent advances in random tensor theory in
the regime of asymptotically large tensor dimensions to analytically
characterize the estimated singular values and the alignment of estimated and
true singular vectors at each step of the deflation procedure. Furthermore,
this result can be used to construct estimators of the signal-to-noise ratios
and the alignments between the estimated and true rank-1 signal
components.Comment: Accepted at IEEE CAMSAP 2023. See also companion paper
arXiv:2304.10248 for the symmetric case. arXiv admin note: text overlap with
arXiv:2211.0900
Information-theoretic bounds and phase transitions in clustering, sparse PCA, and submatrix localization
We study the problem of detecting a structured, low-rank signal matrix
corrupted with additive Gaussian noise. This includes clustering in a Gaussian
mixture model, sparse PCA, and submatrix localization. Each of these problems
is conjectured to exhibit a sharp information-theoretic threshold, below which
the signal is too weak for any algorithm to detect. We derive upper and lower
bounds on these thresholds by applying the first and second moment methods to
the likelihood ratio between these "planted models" and null models where the
signal matrix is zero. Our bounds differ by at most a factor of root two when
the rank is large (in the clustering and submatrix localization problems, when
the number of clusters or blocks is large) or the signal matrix is very sparse.
Moreover, our upper bounds show that for each of these problems there is a
significant regime where reliable detection is information- theoretically
possible but where known algorithms such as PCA fail completely, since the
spectrum of the observed matrix is uninformative. This regime is analogous to
the conjectured 'hard but detectable' regime for community detection in sparse
graphs.Comment: For sparse PCA and submatrix localization, we determine the
information-theoretic threshold exactly in the limit where the number of
blocks is large or the signal matrix is very sparse based on a conditional
second moment method, closing the factor of root two gap in the first versio