https://www.cs.rice.edu/~nakhleh/COMP571/Fall05/DatabaseSearch/blast.pdf

Basic Local Alignment Search Tool

The BLAST programs are widely used tools for searching protein and DNA databases for sequence similarities. For protein comparisons, a variety of definitional, algorithmic and statistical refinements described here permits the execution time of the BLAST programs to be decreased substantially while enhancing their sensitivity to weak similarities. A new criterion for triggering the extension of word hits, combined with a new heuristic for generating gapped alignments, yields a gapped BLAST program that runs at approximately three times the speed of the original. In addition, a method is introduced for automatically combining statistically significant alignments produced by BLAST into a position-specific score matrix, and searching the database using this matrix. The resulting Position-Specific Iterated BLAST (PSIBLAST) program runs at approximately the same speed per iteration as gapped BLAST, but in many cases is much more sensitive to weak but biologically relevant sequence similarities. PSI-BLAST is used to uncover several new and interesting members of the BRCT superfamily.

/pdf/gapped-blast-and-psi-blast-a-new-generation-of-protein-2ybl67x5yx.pdf

Gapped BLAST and PSI-BLAST: a new generation of protein database search programs.

There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

This tutorial provides an overview of the basic theory of hidden Markov models (HMMs) as originated by L.E. Baum and T. Petrie (1966) and gives practical details on methods of implementation of the theory along with a description of selected applications of the theory to distinct problems in speech recognition. Results from a number of original sources are combined to provide a single source of acquiring the background required to pursue further this area of research. The author first reviews the theory of discrete Markov chains and shows how the concept of hidden states, where the observation is a probabilistic function of the state, can be used effectively. The theory is illustrated with two simple examples, namely coin-tossing, and the classic balls-in-urns system. Three fundamental problems of HMMs are noted and several practical techniques for solving these problems are given. The various types of HMMs that have been studied, including ergodic as well as left-right models, are described. >

/pdf/a-tutorial-on-hidden-markov-models-and-selected-applications-1vlzbxbk68.pdf

A tutorial on hidden Markov models and selected applications in speech recognition

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

High-density associative memories

A new method for designing uniform and nonuniform digital filter banks with a specified composite response is presented. The composite response of the filter bank can be met either exactly or to within a given tolerance. We focus on filter banks in which the individual filters are finite impulse response (FIR) digital filters of possibly nonequal length, although the new method is applicable even to more general structures as well. The new method minimizes the weighted sum of the mean square errors in the response of the individual filters, subject to the composite response specifications. Sufficient conditions for either real, ness or phase linearity of the optimal individual filters are presented. The new weighted minimum mean square error (WMMSE) design method is interpreted from a statistical viewpoint as a maximization of the harmonic mean of the output signal-to-noise ratio (SNR) of the individual filters. The complexity of the new method is analyzed, and the design process is demonstrated via a design example.

WMMSE design of digital filter banks with specified composite response

For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.

Random walks among time increasing conductances: heat kernel estimates

Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the trajectories of averaged observables of $(X_i(t))$, initialized from some $\mu$ independent of $\mathbf J$, are universal, i.e., only depend on the choice of the distribution $\mathbf{J}$ through its first and second moments (assuming e.g., sub-exponential tails). We take a general combinatorial approach to proving universality for dynamical systems with random coefficients, combining a stochastic Taylor expansion with a moment matching-type argument. Concrete settings for which our results imply universality include aging in the spherical SK spin glass, and Langevin dynamics and gradient flows for symmetric and asymmetric Hopfield networks.

Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

In this paper we prove that for suitable design specifications and a wide class of optimization criteria, the optimal complex filter bank with specifications on its composite response is composed of frequency translated versions of a prototype filter. In particular, this holds for the min-max and WMMSE (weighted minimum mean square error) criteria. As a result, a simplified design problem whose solution is an optimal prototype filter is formulated. This prototype is essentially an optimal FIR low-pass filter subject to linear constraints on its impulse response. For the WMMSE criterion, this characterization of the optimal filter bank results in a simplified version of the design method presented in [1]. For the min-max criterion, this characterization implies that there exists an optimal window, by which the window design method results in the optimal low-pass prototype. The optimal window design problem is formulated as a linear programming problem, and an approximate solution is derived using the Remez exchange algorithm. For real filter banks in which each filter is composed of a pair of complex filters, the optimal filter bank is no longer composed of frequency translated versions of prototype filter. However, for efficient implementation, the prototype translation property may be part of the design specifications. For this reason, the optimal WMMSE prototype for a class of real filter banks is derived as well.

Amir Dembo

Papers

High-density associative memories

WMMSE design of digital filter banks with specified composite response

Random walks among time increasing conductances: heat kernel estimates

Diffusions interacting through a random matrix: universality via stochastic Taylor expansion

The design of optimal uniform filter banks with specified composite response