We study a particular class of cyclic causal models, where each variable is a (possibly nonlinear) function of its parents and additive noise. We prove that the causal graph of such models is generically identifiable in the bivariate, Gaussian-noise case. We also propose a method to learn such models from observational data. In the acyclic case, the method reduces to ordinary regression, but in the more challenging cyclic case, an additional term arises in the loss function, which makes it a special case of nonlinear independent component analysis. We illustrate the proposed method on synthetic data.

Discriminative learning when training and test data belong to different distributions is a challenging and complex task. Often times we have very few or no labeled data from the test or target distribution but may have plenty of labeled data from multiple related sources with different distributions. The difference in distributions may be both in marginal and conditional probabilities. Most of the existing domain adaptation work focuses on the marginal probability distribution difference between the domains, assuming that the conditional probabilities are similar. However in many real world applications, conditional probability distribution differences are as commonplace as marginal probability differences. In this paper we propose a two-stage domain adaptation methodology which combines weighted data from multiple sources based on marginal probability differences (first stage) as well as conditional probability differences (second stage), with the target domain data. The weights for minimizing the marginal probability differences are estimated independently, while the weights for minimizing conditional probability differences are computed simultaneously by exploiting the potential interaction among multiple sources. We also provide a theoretical analysis on the generalization performance of the proposed multi-source domain adaptation formulation using the weighted Rademacher complexity measure. Empirical comparisons with existing state-of-the-art domain adaptation methods using three real-world datasets demonstrate the effectiveness of the proposed approach.

Many species show avoidance reactions in response to looming object approaches. In locusts, the corresponding escape behavior correlates with the activity of the lobula giant movement detector (LGMD) neuron. During an object approach, its firing rate was reported to gradually increase until a peak is reached, and then it declines quickly. The $\eta$-function predicts that the LGMD activity is a product between an exponential function of angular size $\exp(-\Theta)$ and angular velocity $\dot{\Theta}$, and that peak activity is reached before time-to-contact (ttc). The $\eta$-function has become the prevailing LGMD model because it reproduces many experimental observations, and even experimental evidence for the multiplicative operation was reported. Several inconsistencies remain unresolved, though. Here we address these issues with a new model ($\psi$-model), which explicitly connects $\Theta$ and $\dot{\Theta}$ to biophysical quantities. The $\psi$-model avoids biophysical problems associated with implementing $\exp(\cdot)$, implements the multiplicative operation of $\eta$ via divisive inhibition, and explains why activity peaks could occur after ttc. It consistently predicts response features of the LGMD, and provides excellent fits to published experimental data, with goodness of fit measures comparable to corresponding fits with the $\eta$-function.

We consider the minimization of a convex objective function defined on a Hilbert space, which is only available through unbiased estimates of its gradients. This problem includes standard machine learning algorithms such as kernel logistic regression and least-squares regression, and is commonly referred to as a stochastic approximation problem in the operations research community. We provide a non-asymptotic analysis of the convergence of two well-known algorithms, stochastic gradient descent (a.k.a.~Robbins-Monro algorithm) as well as a simple modification where iterates are averaged (a.k.a.~Polyak-Ruppert averaging). Our analysis suggests that a learning rate proportional to the inverse of the number of iterations, while leading to the optimal convergence rate in the strongly convex case, is not robust to the lack of strong convexity or the setting of the proportionality constant. This situation is remedied when using slower decays together with averaging, robustly leading to the optimal rate of convergence. We illustrate our theoretical results with simulations on synthetic and standard datasets.

We consider large matrices of low rank. We address the problem of recovering such matrices when most of the entries are unknown. Matrix completion finds applications in recommender systems. In this setting, the rows of the matrix may correspond to items and the columns may correspond to users. The known entries are the ratings given by users to some items. The aim is to predict the unobserved ratings. This problem is commonly stated in a constrained optimization framework. We follow an approach that exploits the geometry of the low-rank constraint to recast the problem as an unconstrained optimization problem on the Grassmann manifold. We then apply first- and second-order Riemannian trust-region methods to solve it. The cost of each iteration is linear in the number of known entries. Our methods, RTRMC 1 and 2, outperform state-of-the-art algorithms on a wide range of problem instances.

Recently, there has been substantial interest in using large amounts of unlabeled data to learn word representations which can then be used as features in supervised classifiers for NLP tasks. However, most current approaches are slow to train, do not model the context of the word, and lack theoretical grounding. In this paper, we present a new learning method, Low Rank Multi-View Learning (LR-MVL) which uses a fast spectral method to estimate low dimensional context-specific word representations from unlabeled data. These representation features can then be used with any supervised learner. LR-MVL is extremely fast, gives guaranteed convergence to a global optimum, is theoretically elegant, and achieves state-ofthe-art performanceon named entity recognition (NER) and chunking problems.

Graph cut optimization is one of the standard workhorses of image segmentation since for binary random field representations of the image, it gives globally optimal results and there are efficient polynomial time implementations. Often, the random field is applied over a flat partitioning of the image into non-intersecting elements, such as pixels or super-pixels. In the paper we show that if, instead of a flat partitioning, the image is represented by a hierarchical segmentation tree, then the resulting energy combining unary and boundary terms can still be optimized using graph cut (with all the corresponding benefits of global optimality and efficiency). As a result of such inference, the image gets partitioned into a set of segments that may come from different layers of the tree. We apply this formulation, which we call the pylon model, to the task of semantic segmentation where the goal is to separate an image into areas belonging to different semantic classes. The experiments highlight the advantage of inference on a segmentation tree (over a flat partitioning) and demonstrate that the optimization in the pylon model is able to flexibly choose the level of segmentation across the image. Overall, the proposed system has superior segmentation accuracy on several datasets (Graz-02, Stanford background) compared to previously suggested approaches.

Most action potentials in the nervous system take on the form of strong, rapid, and brief voltage deflections known as spikes, in stark contrast to other action potentials, such as in the heart, that are characterized by broad voltage plateaus. We derive the shape of the neuronal action potential from first principles, by postulating that action potential generation is strongly constrained by the brain's need to minimize energy expenditure. For a given height of an action potential, the least energy is consumed when the underlying currents obey the bang-bang principle: the currents giving rise to the spike should be intense, yet short-lived, yielding spikes with sharp onsets and offsets. Energy optimality predicts features in the biophysics that are not per se required for producing the characteristic neuronal action potential: sodium currents should be extraordinarily powerful and inactivate with voltage; both potassium and sodium currents should have kinetics that have a bell-shaped voltage-dependence; and the cooperative action of multiple `gates' should start the flow of current.

We present a novel regularization-based Multitask Learning (MTL) formulation for Structured Output (SO) prediction for the case of hierarchical task relations. Structured output learning often results in difficult inference problems and requires large amounts of training data to obtain accurate models. We propose to use MTL to exploit information available for related structured output learning tasks by means of hierarchical regularization. Due to the combination of example sets, the cost of training models for structured output prediction can easily become infeasible for real world applications. We thus propose an efficient algorithm based on bundle methods to solve the optimization problems resulting from MTL structured output learning. We demonstrate the performance of our approach on gene finding problems from the application domain of computational biology. We show that 1) our proposed solver achieves much faster convergence than previous methods and 2) that the Hierarchical SO-MTL approach clearly outperforms considered non-MTL methods.

An increasing number of experimental studies indicate that perception encodes a posterior probability distribution over possible causes of sensory stimuli, which is used to act close to optimally in the environment. One outstanding difficulty with this hypothesis is that the exact posterior will in general be too complex to be represented directly, and thus neurons will have to represent an approximation of this distribution. Two influential proposals of efficient posterior representation by neural populations are: 1) neural activity represents samples of the underlying distribution, or 2) they represent a parametric representation of a variational approximation of the posterior. We show that these approaches can be combined for an inference scheme that retains the advantages of both: it is able to represent multiple modes and arbitrary correlations, a feature of sampling methods, and it reduces the represented space to regions of high probability mass, a strength of variational approximations. Neurally, the combined method can be interpreted as a feed-forward preselection of the relevant state space, followed by a neural dynamics implementation of Markov Chain Monte Carlo (MCMC) to approximate the posterior over the relevant states. We demonstrate the effectiveness and efficiency of this approach on a sparse coding model. In numerical experiments on artificial data and image patches, we compare the performance of the algorithms to that of exact EM, variational state space selection alone, MCMC alone, and the combined select and sample approach. The select and sample approach integrates the advantages of the sampling and variational approximations, and forms a robust, neurally plausible, and very efficient model of processing and learning in cortical networks. For sparse coding we show applications easily exceeding a thousand observed and a thousand hidden dimensions.