A sufficient condition for a number to be the order of a nonsingular derivation of a Lie algebra

A study of the set N_p of positive integers which occur as orders of nonsingular derivations of finite-dimensional non-nilpotent Lie algebras of characteristic p>0 was initiated by Shalev and continued by the present author. The main goal of this paper is to show the abundance of elements of N_p. Our main result shows that any divisor n of q-1, where q is a power of p, such that $n\ge (p-1)^{1/p} (q-1)^{1-1/(2p)}$, belongs to N_p. This extends its special case for p=2 which was proved in a previous paper by a different method.Comment: 10 pages. This version has been revised according to a referee's suggestions. The additions include a discussion of the (lower) density of the set N_p, and the results of more extensive machine computations. Note that the title has also changed. To appear in Israel J. Mat

Subgraphs and network motifs in geometric networks

Many real-world networks describe systems in which interactions decay with the distance between nodes. Examples include systems constrained in real space such as transportation and communication networks, as well as systems constrained in abstract spaces such as multivariate biological or economic datasets and models of social networks. These networks often display network motifs: subgraphs that recur in the network much more often than in randomized networks. To understand the origin of the network motifs in these networks, it is important to study the subgraphs and network motifs that arise solely from geometric constraints. To address this, we analyze geometric network models, in which nodes are arranged on a lattice and edges are formed with a probability that decays with the distance between nodes. We present analytical solutions for the numbers of all 3 and 4-node subgraphs, in both directed and non-directed geometric networks. We also analyze geometric networks with arbitrary degree sequences, and models with a field that biases for directed edges in one direction. Scaling rules for scaling of subgraph numbers with system size, lattice dimension and interaction range are given. Several invariant measures are found, such as the ratio of feedback and feed-forward loops, which do not depend on system size, dimension or connectivity function. We find that network motifs in many real-world networks, including social networks and neuronal networks, are not captured solely by these geometric models. This is in line with recent evidence that biological network motifs were selected as basic circuit elements with defined information-processing functions.Comment: 9 pages, 6 figure

Contextual Object Detection with a Few Relevant Neighbors

A natural way to improve the detection of objects is to consider the contextual constraints imposed by the detection of additional objects in a given scene. In this work, we exploit the spatial relations between objects in order to improve detection capacity, as well as analyze various properties of the contextual object detection problem. To precisely calculate context-based probabilities of objects, we developed a model that examines the interactions between objects in an exact probabilistic setting, in contrast to previous methods that typically utilize approximations based on pairwise interactions. Such a scheme is facilitated by the realistic assumption that the existence of an object in any given location is influenced by only few informative locations in space. Based on this assumption, we suggest a method for identifying these relevant locations and integrating them into a mostly exact calculation of probability based on their raw detector responses. This scheme is shown to improve detection results and provides unique insights about the process of contextual inference for object detection. We show that it is generally difficult to learn that a particular object reduces the probability of another, and that in cases when the context and detector strongly disagree this learning becomes virtually impossible for the purposes of improving the results of an object detector. Finally, we demonstrate improved detection results through use of our approach as applied to the PASCAL VOC and COCO datasets

Primitive Words, Free Factors and Measure Preservation

Let F_k be the free group on k generators. A word w \in F_k is called primitive if it belongs to some basis of F_k. We investigate two criteria for primitivity, and consider more generally, subgroups of F_k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H \le J \le F_k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F_k is primitive. Again let w \in F_k and consider the word map w:G x G x ... x G \to G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G x G x ... x G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and in particular prove the conjecture for k=2. It was asked whether the primitive elements of F_k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F_2.Comment: This is a unified version of two manuscripts: "On Primitive words I: A New Algorithm", and "On Primitive Words II: Measure Preservation". 42 pages, 14 figures. Some parts of the paper reorganized towards publication in the Israel J. of Mat

Generalization Error in Deep Learning

Deep learning models have lately shown great performance in various fields such as computer vision, speech recognition, speech translation, and natural language processing. However, alongside their state-of-the-art performance, it is still generally unclear what is the source of their generalization ability. Thus, an important question is what makes deep neural networks able to generalize well from the training set to new data. In this article, we provide an overview of the existing theory and bounds for the characterization of the generalization error of deep neural networks, combining both classical and more recent theoretical and empirical results

Scalable and Interpretable One-class SVMs with Deep Learning and Random Fourier features

One-class support vector machine (OC-SVM) for a long time has been one of the most effective anomaly detection methods and extensively adopted in both research as well as industrial applications. The biggest issue for OC-SVM is yet the capability to operate with large and high-dimensional datasets due to optimization complexity. Those problems might be mitigated via dimensionality reduction techniques such as manifold learning or autoencoder. However, previous work often treats representation learning and anomaly prediction separately. In this paper, we propose autoencoder based one-class support vector machine (AE-1SVM) that brings OC-SVM, with the aid of random Fourier features to approximate the radial basis kernel, into deep learning context by combining it with a representation learning architecture and jointly exploit stochastic gradient descent to obtain end-to-end training. Interestingly, this also opens up the possible use of gradient-based attribution methods to explain the decision making for anomaly detection, which has ever been challenging as a result of the implicit mappings between the input space and the kernel space. To the best of our knowledge, this is the first work to study the interpretability of deep learning in anomaly detection. We evaluate our method on a wide range of unsupervised anomaly detection tasks in which our end-to-end training architecture achieves a performance significantly better than the previous work using separate training.Comment: Accepted at European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML-PKDD) 201

Coarse-Graining and Self-Dissimilarity of Complex Networks

Can complex engineered and biological networks be coarse-grained into smaller and more understandable versions in which each node represents an entire pattern in the original network? To address this, we define coarse-graining units (CGU) as connectivity patterns which can serve as the nodes of a coarse-grained network, and present algorithms to detect them. We use this approach to systematically reverse-engineer electronic circuits, forming understandable high-level maps from incomprehensible transistor wiring: first, a coarse-grained version in which each node is a gate made of several transistors is established. Then, the coarse-grained network is itself coarse-grained, resulting in a high-level blueprint in which each node is a circuit-module made of multiple gates. We apply our approach also to a mammalian protein-signaling network, to find a simplified coarse-grained network with three main signaling channels that correspond to cross-interacting MAP-kinase cascades. We find that both biological and electronic networks are 'self-dissimilar', with different network motifs found at each level. The present approach can be used to simplify a wide variety of directed and nondirected, natural and designed networks.Comment: 11 pages, 11 figure

Coordinate Descent with Arbitrary Sampling I: Algorithms and Complexity

postprin

Premise Selection for Mathematics by Corpus Analysis and Kernel Methods

Smart premise selection is essential when using automated reasoning as a tool for large-theory formal proof development. A good method for premise selection in complex mathematical libraries is the application of machine learning to large corpora of proofs. This work develops learning-based premise selection in two ways. First, a newly available minimal dependency analysis of existing high-level formal mathematical proofs is used to build a large knowledge base of proof dependencies, providing precise data for ATP-based re-verification and for training premise selection algorithms. Second, a new machine learning algorithm for premise selection based on kernel methods is proposed and implemented. To evaluate the impact of both techniques, a benchmark consisting of 2078 large-theory mathematical problems is constructed,extending the older MPTP Challenge benchmark. The combined effect of the techniques results in a 50% improvement on the benchmark over the Vampire/SInE state-of-the-art system for automated reasoning in large theories.Comment: 26 page

Male frequent attenders of general practice and their help seeking preferences

Background: Low rates of health service usage by men are commonly linked to masculine values and traditional male gender roles. However, not all men conform to these stereotypical notions of masculinity, with some men choosing to attend health services on a frequent basis, for a variety of different reasons. This study draws upon the accounts of male frequent attenders of the General Practitioner's (GP) surgery, examining their help-seeking preferences and their reasons for choosing services within general practice over other sources of support. Methods: The study extends thematic analysis of interview data from the Self Care in Primary Care study (SCinPC), a large scale multi-method evaluation study of a self care programme delivered to frequent attenders of general practice. Data were collected from 34 semi-structured interviews conducted with men prior to their exposure to the intervention. Results: The ages of interviewed men ranged from 16 to 72 years, and 91% of the sample (n= 31) stated that they had a current health condition. The thematic analysis exposed diverse perspectives within male help-seeking preferences and the decision-making behind men's choice of services. The study also draws attention to the large variation in men's knowledge of available health services, particularly alternatives to general practice. Furthermore, the data revealed some men's lack of confidence in existing alternatives to general practice. Conclusions: The study highlights the complex nature of male help-seeking preferences, and provides evidence that there should be no 'one size fits all' approach to male service provision. It also provides impetus for conducting further studies into this under researched area of interest. © 2011 WPMH GmbH