27,181 research outputs found
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
Distance-Biregular Graphs and Orthogonal Polynomials
This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems.
We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new
results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound
An estimate for the thermal photon rate from lattice QCD
We estimate the production rate of photons by the quark-gluon plasma in
lattice QCD. We propose a new correlation function which provides better
control over the systematic uncertainty in estimating the photon production
rate at photon momenta in the range {\pi}T/2 to 2{\pi}T. The relevant Euclidean
vector current correlation functions are computed with = 2
Wilson clover fermions in the chirally-symmetric phase. In order to estimate
the photon rate, an ill-posed problem for the vector-channel spectral function
must be regularized. We use both a direct model for the spectral function and a
model-independent estimate from the Backus-Gilbert method to give an estimate
for the photon rate.Comment: 15 pages, 11 figures, talk presented at 35th annual International
Symposium on Lattice Field Theory, 18-24 June 2017, Granada, Spai
On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters
Kalman filter is a key tool for time-series forecasting and analysis. We show
that the dependence of a prediction of Kalman filter on the past is decaying
exponentially, whenever the process noise is non-degenerate. Therefore, Kalman
filter may be approximated by regression on a few recent observations.
Surprisingly, we also show that having some process noise is essential for the
exponential decay. With no process noise, it may happen that the forecast
depends on all of the past uniformly, which makes forecasting more difficult.
Based on this insight, we devise an on-line algorithm for improper learning
of a linear dynamical system (LDS), which considers only a few most recent
observations. We use our decay results to provide the first regret bounds
w.r.t. to Kalman filters within learning an LDS. That is, we compare the
results of our algorithm to the best, in hindsight, Kalman filter for a given
signal. Also, the algorithm is practical: its per-update run-time is linear in
the regression depth
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