Nonnegative matrix factorization (NMF) approximates a given data matrix as a product of two low-rank nonnegative matrices, usually by minimizing the L2 or the KL distance between the data matrix and the matrix product. This factorization was shown to be useful for several important computer vision applications. We propose here two new NMF algorithms that minimize the Earth mover's distance (EMD) error between the data and the matrix product. The algorithms (EMD NMF and bilateral EMD NMF) are iterative and based on linear programming methods. We prove their convergence, discuss their numerical difficulties, and propose efficient approximations. Naturally, the matrices obtained with EMD NMF are different from those obtained with L2-NMF. We discuss these differences in the context of two challenging computer vision tasks, texture classification and face recognition, perform actual NMF-based image segmentation for the first time, and demonstrate the advantages of the new methods with common benchmarks.
Receptive field identification is a vital problem in sensory neurophysiology and vision. Much research has been done in identifying the receptive fields of nonlinear neurons whose firing rate is determined by the nonlinear interactions of a small number of linear filters. Despite more advanced methods that have been proposed, spike-triggered covariance (STC) continues to be the most widely used method in such situations due to its simplicity and intuitiveness. Although the connection between STC and Wiener/Volterra kernels has often been mentioned in the literature, this relationship has never been explicitly derived. Here we derive this relationship and show that the STC matrix is actually a modified version of the second-order Wiener kernel, which incorporates the input autocorrelation and mixes first- and second-order dynamics. It is then shown how, with little modification of the STC method, the Wiener kernels may be obtained and, from them, the principal dynamic modes, a set of compact and efficient linear filters that essentially combine the spike-triggered average and STC matrix and generalize to systems with both continuous and point-process outputs. Finally, using Wiener theory, we show how these obtained filters may be corrected when they were estimated using correlated inputs. Our correction technique is shown to be superior to those commonly used in the literature for both correlated Gaussian images and natural images.
Although an anatomical connection from CA1 to CA3 via the Entorhinal Cortex (EC) and through backprojecting interneurons has long been known it exist, it has never been examined quantitatively on the single neuron level, in the in-vivo nonpatholgical, nonperturbed brain. Here, single spike activity was recorded using a multi-electrode array from the CA3 and CA1 areas of the rodent hippocampus (N=7) during a behavioral task. The predictive power from CA3→CA1 and CA1→CA3 was examined by constructing Multivariate Autoregressive (MVAR) models from recorded neurons in both directions. All nonsignificant inputs and models were identified and removed by means of Monte Carlo simulation methods. It was found that 121/166 (73%) CA3→CA1 models and 96/145 (66%) CA1→CA3 models had significant predictive power, thus confirming a predictive ‘Granger’ causal relationship from CA1 to CA3. This relationship is thought to be caused by a combination of truly causal connections such as the CA1→EC→CA3 pathway and common inputs such as those from the Septum. All MVAR models were then examined in the frequency domain and it was found that CA3 kernels had significantly more power in the theta and beta range than those of CA1, confirming CA3’s role as an endogenous hippocampal pacemaker.
Most email applications devote a significant part of their real estate to organization mechanisms such as folders. Yet, we verified on the Yahoo! Mail service that 70% of email users have never defined a single folder. This implies that one of the most well known email features is underexploited. We propose here to revive the feature by providing a method for generating a lighter form of folders, or tags, benefiting even the most passive users. The method automatically associates, whenever possible, an appropriate semantic tag with a given email. This gives rise to an alternate mechanism for organizing and searching email.We advocate a novel modeling approach that exploits the overall population of users, thereby learning from the wisdomof-crowds how to categorize messages. Given our massive user base, it is enough to learn from a minority of the users who label certain messages in order to label that kind of messages for the general population. We design a novel cascade classification approach, which copes with the severe scalability and accuracy constraints we are facing. Significant efficiency gains are achieved by working within a low dimensional latent space, and by using a novel hierarchical classifier. Precision level is controlled by separating the task into a two-phase classification process.We performed an extensive empirical study covering three different time periods, over 100 million messages, and thousands of candidate tags per message. The results are encouraging and compare favorably with alternative approaches. Our method successfully tags 72% of incoming email traffic. Performance-wise, the computational overhead, even on surge large traffic, is sufficiently low for our approach to be applicable in production on any large Web mail service.
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