We revisit the idea of brain damage, i.e. the pruning of the coefficients of a neural network, and suggest how brain damage can be modified and used to speedup convolutional layers. The approach uses the fact that many efficient implementations reduce generalized convolutions to matrix multiplications. The suggested brain damage process prunes the convolutional kernel tensor in a group-wise fashion by adding group-sparsity regularization to the standard training process. After such group-wise pruning, convolutions can be reduced to multiplications of thinned dense matrices, which leads to speedup. In the comparison on AlexNet, the method achieves very competitive performance.
Our approach is also related to methods that use structured sparsity @cite_44 @cite_37 @cite_23 to discover optimal architectures of certain machine learners, e.g. to discover the optimal structure of a graphical model @cite_18 or the optimal receptive fields in the two layered image classifier @cite_21 . On the other hand, since our approach effectively learns receptive fields within a ConvNet, it can be related to other receptive field learning approaches, e.g. @cite_0 @cite_40 .
@cite_37 @cite_18 @cite_21 @cite_44 @cite_0 @cite_40 @cite_23
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The Group-Lasso method for finding important explanatory factors suffers from the potential non-uniqueness of solutions and also from high computational costs. We formulate conditions for the uniqueness of Group-Lasso solutions which lead to an easily implementable test procedure that allows us to identify all potentially active groups. These results are used to derive an efficient algorithm that can deal with input dimensions in the millions and can approximate the solution path efficiently. The derived methods are applied to large-scale learning problems where they exhibit excellent performance and where the testing procedure helps to avoid misinterpretations of the solutions. We study the problem of learning the graph structure associated with a general discrete graphical models (each variable can take any of m > 1 values, the clique factors have maximum size c ≥ 2) from samples, under high-dimensional scaling where the number of variables p could be larger than the number of samples n. We provide a quantitative consistency analysis of a procedure based on node-wise multi-class logistic regression with group-sparse regularization. We first consider general m-ary pairwise models – where each factor depends on at most two variables. We show that when the number of samples scale as n > K(m − 1) 2 d 2 log((m −1) 2 (p −1))– where d is the maximum degree and K a fixed constant – the procedure succeeds in recovering the graph with high probability. For general models with c-way factors, the natural multi-way extension of the pairwise method quickly becomes very computationally complex. So we studied the effectiveness of using the pairwise method even while the true model has higher order factors. Surprisingly, we show that under slightly more stringent conditions, the pairwise procedure still recovers the graph structure, when the samples scale as n > K(m − 1) 2 d 3 2 c 1 log((m − 1) c (p − 1) c 1 ). Summary. We consider the problem of selecting grouped variables (factors) for accurate prediction in regression. Such a problem arises naturally in many practical situations with the multifactor analysis-of-variance problem as the most important and well-known example. Instead of selecting factors by stepwise backward elimination, we focus on the accuracy of estimation and consider extensions of the lasso, the LARS algorithm and the non-negative garrotte for factor selection. The lasso, the LARS algorithm and the non-negative garrotte are recently proposed regression methods that can be used to select individual variables. We study and propose efficient algorithms for the extensions of these methods for factor selection and show that these extensions give superior performance to the traditional stepwise backward elimination method in factor selection problems. We study the similarities and the differences between these methods. Simulations and real examples are used to illustrate the methods. From the early HMAX model to Spatial Pyramid Matching, spatial pooling has played an important role in visual recognition pipelines. By aggregating local statistics, it equips the recognition pipelines with a certain degree of robustness to translation and deformation yet preserving spatial information. Despite of its predominance in current recognition systems, we have seen little progress to fully adapt the pooling strategy to the task at hand. In this paper, we propose a flexible parameterization of the spatial pooling step and learn the pooling regions together with the classifier. We investigate a smoothness regularization term that in conjuncture with an efficient learning scheme makes learning scalable. Our framework can work with both popular pooling operators: sum-pooling and max-pooling. Finally, we show benefits of our approach for object recognition tasks based on visual words and higher level event recognition tasks based on object-bank features. In both cases, we improve over the hand-crafted spatial pooling step showing the importance of its adaptation to the task. We consider the empirical risk minimization problem for linear supervised learning, with regularization by structured sparsity-inducing norms. These are defined as sums of Euclidean norms on certain subsets of variables, extending the usual l1-norm and the group l1-norm by allowing the subsets to overlap. This leads to a specific set of allowed nonzero patterns for the solutions of such problems. We first explore the relationship between the groups defining the norm and the resulting nonzero patterns, providing both forward and backward algorithms to go back and forth from groups to patterns. This allows the design of norms adapted to specific prior knowledge expressed in terms of nonzero patterns. We also present an efficient active set algorithm, and analyze the consistency of variable selection for least-squares linear regression in low and high-dimensional settings.