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

We study dynamics for asymmetric spin glass models, proposed by Hertz et al. and Sompolinsky et al. in the 1980's in the context of neural networks: particles evolve via a modified Langevin dynamics for the Sherrington--Kirkpatrick model with soft spins, whereby the disorder is i.i.d. standard Gaussian rather than symmetric. Ben Arous and Guionnet (1995), followed by Guionnet (1997), proved for Gaussian interactions that as the number of particles grows, the short-term empirical law of this dynamics converges a.s. to a non-random law $\mu_\star$ of a ``self-consistent single spin dynamics,'' as predicted by physicists. Here we obtain universality of this fact: 
For asymmetric disorder given by i.i.d. variables of zero mean, unit variance and exponential or better tail decay, at every temperature, the empirical law of sample paths of the 
Langevin-like dynamics in a fixed time interval has the same a.s. limit $\mu_\star$.

Universality for Langevin-like spin glass dynamics

The semi-infinite Atlas process is a one-dimensional system of Brownian particles, where only the leftmost particle gets a unit drift to the right. Its particle spacing process has infinitely many stationary measures, with one distinguished translation invariant reversible measure. We show that the latter is attractive for a large class of initial configurations of slowly growing (or bounded) particle densities. Key to our proof is a new estimate on the rate of convergence to equilibrium for the particle spacing in a triangular array of finite, large size systems.

The infinite Atlas process: Convergence to equilibrium.

Let $T_n$ denote the binary tree of depth $n$ augmented by an extra edge connected to its root. Let $C_n$ denote the cover time of $T_n$ by simple random walk. We prove that $\sqrt{ \mathcal{C}_{n} 2^{-(n+1) } } - m_n$ converges in distribution as $n\to \infty$, where $m_n$ is an explicit constant, and identify the limit.

Limit law for the cover time of a random walk on a binary tree

A rigorous analysis on the finite precision computational aspects of neural network as a pattern classifier via a probabilistic approach is presented. Even though there exist negative results on the capability of perceptron, we show the following positive results: Given n pattern vectors each represented by cn bits where c > 1, that are uniformly distributed, with high probability the perceptron can perform all possible binary classifications of the patterns. Moreover, the resulting neural network requires a vanishingly small proportion O(log n/n) of the memory that would be required for complete storage of the patterns. Further, the perceptron algorithm takes O(n2) arithmetic operations with high probability, whereas other methods such as linear programming takes O(n3.5) in the worst case. We also indicate some mathematical connections with VLSI circuit testing and the theory of random matrices.

/pdf/complexity-of-finite-precision-neural-network-classifier-32p64c1xji.pdf

Complexity of Finite Precision Neural Network Classifier

We consider recurrence versus transience for models of random walks on growing in time, connected subsets $\mathbb{G}_t$ of some fixed locally finite, connected graph, in which monotone interaction enforces such growth as a result of visits by the walk (or probes it sent), to the neighborhood of the boundary of $\mathbb{G}_t$.

/pdf/monotone-interaction-of-walk-and-graph-recurrence-versus-4pih1c8glw.pdf

Amir Dembo

Papers

Universality for Langevin-like spin glass dynamics

The infinite Atlas process: Convergence to equilibrium.

Limit law for the cover time of a random walk on a binary tree

Complexity of Finite Precision Neural Network Classifier

Monotone interaction of walk and graph: recurrence versus transience