Deep learning is a form of machine learning that enables computers to learn from experience and understand the world in terms of a hierarchy of concepts. Because the computer gathers knowledge from experience, there is no need for a human computer operator to formally specify all the knowledge that the computer needs. The hierarchy of concepts allows the computer to learn complicated concepts by building them out of simpler ones; a graph of these hierarchies would be many layers deep. This book introduces a broad range of topics in deep learning. The text offers mathematical and conceptual background, covering relevant concepts in linear algebra, probability theory and information theory, numerical computation, and machine learning. It describes deep learning techniques used by practitioners in industry, including deep feedforward networks, regularization, optimization algorithms, convolutional networks, sequence modeling, and practical methodology; and it surveys such applications as natural language processing, speech recognition, computer vision, online recommendation systems, bioinformatics, and videogames. Finally, the book offers research perspectives, covering such theoretical topics as linear factor models, autoencoders, representation learning, structured probabilistic models, Monte Carlo methods, the partition function, approximate inference, and deep generative models. Deep Learning can be used by undergraduate or graduate students planning careers in either industry or research, and by software engineers who want to begin using deep learning in their products or platforms. A website offers supplementary material for both readers and instructors.

Deep Learning

From the Publisher:
A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols; more than 200 tables and figures; more than 1,000 numbered definitions, facts, examples, notes, and remarks; and over 1,250 significant references, including brief comments on each paper.

Handbook of Applied Cryptography

The success of machine learning algorithms generally depends on data representation, and we hypothesize that this is because different representations can entangle and hide more or less the different explanatory factors of variation behind the data. Although specific domain knowledge can be used to help design representations, learning with generic priors can also be used, and the quest for AI is motivating the design of more powerful representation-learning algorithms implementing such priors. This paper reviews recent work in the area of unsupervised feature learning and deep learning, covering advances in probabilistic models, autoencoders, manifold learning, and deep networks. This motivates longer term unanswered questions about the appropriate objectives for learning good representations, for computing representations (i.e., inference), and the geometrical connections between representation learning, density estimation, and manifold learning.

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Representation Learning: A Review and New Perspectives

This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP/PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.

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Tensor Decompositions and Applications

Can machine learning deliver AI? Theoretical results, inspiration from the brain and cognition, as well as machine learning experiments suggest that in order to learn the kind of complicated functions that can represent high-level abstractions (e.g. in vision, language, and other AI-level tasks), one would need deep architectures. Deep architectures are composed of multiple levels of non-linear operations, such as in neural nets with many hidden layers, graphical models with many levels of latent variables, or in complicated propositional formulae re-using many sub-formulae. Each level of the architecture represents features at a different level of abstraction, defined as a composition of lower-level features. Searching the parameter space of deep architectures is a difficult task, but new algorithms have been discovered and a new sub-area has emerged in the machine learning community since 2006, following these discoveries. Learning algorithms such as those for Deep Belief Networks and other related unsupervised learning algorithms have recently been proposed to train deep architectures, yielding exciting results and beating the state-of-the-art in certain areas. Learning Deep Architectures for AI discusses the motivations for and principles of learning algorithms for deep architectures. By analyzing and comparing recent results with different learning algorithms for deep architectures, explanations for their success are proposed and discussed, highlighting challenges and suggesting avenues for future explorations in this area.

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Learning Deep Architectures for AI

A hierarchy of probabilistic complexity classes generalizing NP has recently emerged in the work of [B], [GMR], and [GS]. The IP hierarchy is defined through the notion of an interactive proof system, in which an all powerful prover tries to convince a probabilistic polynomial time verifier that a string x is in a language L. The verifier tosses coins and exchanges messages back and forth with the prover before he decides whether to accept x. This proof-system yields "probabilistic" proofs: the verifier may erroneously accept or reject x with small probability. The class IP[f(|x|)] is said to contain L if, there exists an interactive proof system with f(|x|)- message exchanges (interactions) such that with high probability the verifier accepts x if and only if x e L. Babai [B] showed that all languages recognized by interactive proof systems with bounded number of interactions, can be recognized by interactive proof systems with only two interactions. Namely, for every constant k, IP[k] collapses to Ip[2]. In this paper, we give evidence that interactive proof systems with unbounded number of interactions may be more powerful than interactive proof systems with bounded number of interactions. We show that for any unbounded function f(n) there exists an oracle B such that IPB [f(|x|)] ⊄ PHB. This implies that IPB[f(n)] ≠ IPB[2], since IPB[2] ⊆ Π2B for all oracles B. The techniques employed are extensions of the techniques for proving lower bounds on small depth circuits used in [FSS], [Y] and [H1].

On the power of interaction

The authors consider the following NP optimization problem: given a set of polynomials P/sub i/(x), i=1. . .s of degree at most 2 over GF(p) in n variables, find a root common to as many as possible of the polynomials P/sub i/(x). They prove that in the case when the polynomials do not contain any squares as monomials, it is always possible to approximate this problem within a factor of /sup p2///sub p-1/ in polynomial time. This follows from the stronger statement that one can, in polynomial time, find an assignment that satisfies at least /sup p-1///sub p2/ of the nontrivial equations. More interestingly, they prove that approximating the maximal number of polynomials with a common root to within a factor of p- in is NP-hard. They also prove that for any constant delta >

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A well-characterized approximation problem

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The square lattice shuffle

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Theta(√(n)) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.

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Towards an optimal separation of space and length in resolution

https://www.csc.kth.se/%7Ejohanh/lmnsharp.pdf

Johan Håstad

Papers

On the power of interaction

A well-characterized approximation problem

The square lattice shuffle

Towards an optimal separation of space and length in resolution

A Slight Sharpening of LMN