In numerous signal processing and machine learning applications, the problem of signal recovery from a limited number of nonlinear observations is of special interest. These problems also called inverse problem have r...
详细信息
In numerous signal processing and machine learning applications, the problem of signal recovery from a limited number of nonlinear observations is of special interest. These problems also called inverse problem have recently received attention in signal processing, machine learning, and high-dimensional statistics. In high-dimensional setting, the inverse problems are inherently ill-posed as the number of measurements is typically less than the number of dimensions. As a result, one needs to assume some structures on the underlying signal such as sparsity, structured sparsity, low-rank and so on. In addition, having a nonlinear map from the signal space to the measurement space may add more challenges to the problem. For instance, the assumption on the nonlinear function such as known/unknown, invertibility, smoothness, even/odd, and so on can change the tractability of the problem dramatically. The nonlinear inverse problems are also a special interest in the context of neural network and deep learning as each layer can be cast as an instance of the inverse problem. As a result, understanding of an inverse problem can serve as a building block for more general and complex networks. In this thesis, we study various aspects of such inverse problems with focusing on the underlying signal structure, the compression modes, the nonlinear map from signal space to measurement space, and the connection of the inverse problems to the analysis of some class of neural networks. In this regard, we try to answer statistical properties and computational limits of the proposed methods, and compare them to the state-of-the-art approaches. First, we start with the superposition signal model in which the underlying signal is assumed to be the superposition of two components with sparse representation (i.e., their support is arbitrary sparse) in some specific domains. Initially, we assume that the nonlinear function also called link function is not known. Then, the goal is defined as
Sparse coding is a crucial subroutine in algorithms for various signal processing, deep learning, and other machine learning applications. The central goal is to learn an overcomplete dictionary that can sparsely repr...
详细信息
Sparse coding is a crucial subroutine in algorithms for various signal processing, deep learning, and other machine learning applications. The central goal is to learn an overcomplete dictionary that can sparsely represent a given input dataset. However, a key challenge is that storage, transmission, and processing of the learned dictionary can be untenably high if the data dimension is high. In this paper, we consider the double-sparsity model introduced by Rubinstein et al. (2010b) where the dictionary itself is the product of a fixed, known basis and a data-adaptive sparse component. First, we introduce a simple algorithm for double-sparse coding that can be amenable to efficient implementation via neural architectures. Second, we theoretically analyze its performance and demonstrate asymptotic sample complexity and running time benefits over existing (provable) approaches for sparse coding. To our knowledge, our work introduces the first computationally efficient algorithm for double-sparse coding that enjoys rigorous statistical guarantees. Finally, we corroborate our theory with several numerical experiments on simulated data, suggesting that our method may be useful for problem sizes encountered in practice.
The problem of path planning for an automaton moving in a two-dimensional scene filled with unknown obstacles is considered. The automaton is presented as a point; obstacles can be of an arbitrary shape, with continuo...
详细信息
The problem of path planning for an automaton moving in a two-dimensional scene filled with unknown obstacles is considered. The automaton is presented as a point; obstacles can be of an arbitrary shape, with continuous boundaries and of finite size; no restriction on the size of the scene is imposed. The information available to the automaton is limited to its own current coordinates and those of the target position. Also, when the automaton hits an obstacle, this fact is detected by the automaton's “tactile sensor.” This information is shown to be sufficient for reaching the target or concluding in finite time that the target cannot be reached. A worst-case lower bound on the length of paths generated by any algorithm operating within the framework of the accepted model is developed; the bound is expressed in terms of the perimeters of the obstacles met by the automaton in the scene. algorithms that guarantee reaching the target (if the target is reachable), and tests for target reachability are presented. The efficiency of the algorithms is studied, and worst-case upper bounds on the length of generated paths are produced.
In the nonnegative matrix factorization (NMF) problem we are given an n x m nonnegative matrix M and an integer r > 0. Our goal is to express M as AW, where A and W are nonnegative matrices of size n x r and r x m,...
详细信息
In the nonnegative matrix factorization (NMF) problem we are given an n x m nonnegative matrix M and an integer r > 0. Our goal is to express M as AW, where A and W are nonnegative matrices of size n x r and r x m, respectively. In some applications, it makes sense to ask instead for the product AW to approximate M, i.e. (approximately) minimize parallel to M - AW(F) parallel to, where parallel to parallel to(F), denotes the Frobenius norm;we refer to this as approximate NMF. This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where A and W are computed using a variety of local search heuristics. Vavasis recently proved that this problem is NP-complete. (Without the restriction that A and W be nonnegative, both the exact and approximate problems can be solved optimally via the singular value decomposition.) We initiate a study of when this problem is solvable in polynomial time. Our results are the following: 1. We give a polynomial-time algorithm for exact and approximate NMF for every constant r. Indeed NMF is most interesting in applications precisely when r is small. 2. We complement this with a hardness result, that if exact NMF can be solved in time (nm)(o(r)), 3-SAT has a subexponential-time algorithm. This rules out substantial improvements to the above algorithm. 3. We give an algorithm that runs in time polynomial in n, m, and r under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings. To the best of our knowledge, this last result is the first example of a polynomial-time algorithm that provably works under a non-trivial condition on the input and we believe that this will be an interesti
Proteins are essential components of cells and are crucial for catalyzing reactions, signaling, recognition, motility, recycling, and structural stability. This diversity of function suggests that nature is only scrat...
详细信息
Proteins are essential components of cells and are crucial for catalyzing reactions, signaling, recognition, motility, recycling, and structural stability. This diversity of function suggests that nature is only scratching the surface of protein functional space. Protein function is determined by structure, which in turn is determined predominantly by amino acid sequence. Protein design aims to explore protein sequence and conformational space to design novel proteins with new or improved function. The vast number of possible protein sequences makes exploring the space a challenging problem. Computational structure-based protein design (CSPD) allows for the rational design of proteins. Because of the large search space, CSPD methods must balance search accuracy and modeling simplifications. We have developed algorithms that allow for the accurate and efficient search of protein conformational space. Specifically, we focus on algorithms that maintain provability, account for protein flexibility, and use ensemble-based rankings. We present several novel algorithms for incorporating improved flexibility into CSPD with continuous rotamers. We applied these algorithms to two biomedically important design problems. We designed peptide inhibitors of the cystic fibrosis agonist CAL that were able to restore function of the vital cystic fibrosis protein CFTR. We also designed improved HIV antibodies and nanobodies to combat HIV infections.
Broadly neutralizing antibodies (bNAbs) against HIV can reduce viral transmission in humans, but an effective therapeutic will require unusually high breadth and potency of neutralization. We employ the OSPREY computa...
详细信息
Broadly neutralizing antibodies (bNAbs) against HIV can reduce viral transmission in humans, but an effective therapeutic will require unusually high breadth and potency of neutralization. We employ the OSPREY computational protein design software to engineer variants of two apex-directed bNAbs, PGT145 and PG9RSH, resulting in increases in potency of over 100-fold against some viruses. The top designed variants improve neutralization breadth from 39% to 54% at clinically relevant concentrations (IC80 < 1 mg/mL) and improve median potency (IC80) by up to 4-fold over a cross-clade panel of 208 strains. To investigate the mechanisms of improvement, we determine cryoelectron microscopy structures of each variant in complex with the HIV envelope trimer. Surprisingly, we find the largest increases in breadth to be a result of optimizing side-chain interactions with highly variable epitope residues. These results provide insight into mechanisms of neutralization breadth and inform strategies for antibody design and improvement.
暂无评论