We present the first provably sublinear time hashing algorithm for approximate \emph{Maximum Inner Product Search} (MIPS). Searching with (un-normalized) inner product as the underlying similarity measure is a known difficult problem and finding hashing schemes for MIPS was considered hard. While the existing Locality Sensitive Hashing (LSH) framework is insufficient for solving MIPS, in this paper we extend the LSH framework to allow asymmetric hashing schemes. Our proposal is based on a key observation that the problem of finding maximum inner products, after independent asymmetric transformations, can be converted into the problem of approximate near neighbor search in classical settings. This key observation makes efficient sublinear hashing scheme for MIPS possible. Under the extended asymmetric LSH (ALSH) framework, this paper provides an example of explicit construction of provably fast hashing scheme for MIPS. Our proposed algorithm is simple and easy to implement. The proposed hashing scheme leads to significant computational savings over the two popular conventional LSH schemes: (i) Sign Random Projection (SRP) and (ii) hashing based on $p$-stable distributions for $L_2$ norm (L2LSH), in the collaborative filtering task of item recommendations on Netflix and Movielens (10M) datasets.

Estimating a vector from noisy quadratic observations is a task that arises naturally in many contexts, from dimensionality reduction, to synchronization and phase retrieval problems. It is often the case that additional information is available about the unknown vector (for instance, sparsity, sign or magnitude of its entries). Many authors propose non-convex quadratic optimization problems that aim at exploiting optimally this information. However, solving these problems is typically NP-hard. We consider a simple model for noisy quadratic observation of an unknown vector $\bvz$. The unknown vector is constrained to belong to a cone $\Cone \ni \bvz$. While optimal estimation appears to be intractable for the general problems in this class, we provide evidence that it is tractable when $\Cone$ is a convex cone with an efficient projection. This is surprising, since the corresponding optimization problem is non-convex and --from a worst case perspective-- often NP hard. We characterize the resulting minimax risk in terms of the statistical dimension of the cone $\delta(\Cone)$. This quantity is already known to control the risk of estimation from gaussian observations and random linear measurements. It is rather surprising that the same quantity plays a role in the estimation risk from quadratic measurements.

We present a theoretical analysis of F -measures for binary, multiclass and multilabel classification.These performance measures are nonlinear, but in many scenarios they are pseudo-linear functions of the per-class false negative/false positive rate. Based on this observation, we present a general reduction of F - measure maximization to cost-sensitive classification with unknown costs. We then propose an algorithm with provable guarantees to obtain an approximately optimal classifier for the F -measure by solving a series of cost-sensitive classification problems.The strength of our analysis is to be valid on any dataset and any class of classifiers, extending the existing theoretical results on F -measures, which are asymptotic in nature. We present numerical experiments to illustrate the relative importance of cost asymmetry and thresholding when learning linear classifiers on various F -measure optimization tasks.

One of the most fundamental problems in causal inference is the estimation of a causal effect when variables are confounded. This is difficult in an observational study because one has no direct evidence that all confounders have been adjusted for. We introduce a novel approach for estimating causal effects that exploits observational conditional independencies to suggest ``weak'' paths in a unknown causal graph. The widely used faithfulness condition of Spirtes et al. is relaxed to allow for varying degrees of ``path cancellations'' that will imply conditional independencies but do not rule out the existence of confounding causal paths. The outcome is a posterior distribution over bounds on the average causal effect via a linear programming approach and Bayesian inference. We claim this approach should be used in regular practice to complement other default tools in observational studies.

Deep learning has been a long standing pursuit in machine learning, which until recently was hampered by unreliable training methods before the discovery of improved heuristics for embedded layer training. A complementary research strategy is to develop alternative modeling architectures that admit efficient training methods while expanding the range of representable structures toward deep models. In this paper, we develop a new architecture for nested nonlinearities that allows arbitrarily deep compositions to be trained to global optimality. The approach admits both parametric and nonparametric forms through the use of normalized kernels to represent each latent layer. The outcome is a fully convex formulation that is able to capture compositions of trainable nonlinear layers to arbitrary depth.

Lazy local learning methods train a classifier on the fly" at test time, using only a subset of the training instances that are most relevant to the novel test example. The goal is to tailor the classifier to the properties of the data surrounding the test example. Existing methods assume that the instances most useful for building the local model are strictly those closest to the test example. However, this fails to account for the fact that the success of the resulting classifier depends on the full distribution of selected training instances. Rather than simply gather the test example's nearest neighbors, we propose to predict the subset of training data that is jointly relevant to training its local model. We develop an approach to discover patterns between queries and their "good" neighborhoods using large-scale multi-label classification with compressed sensing. Given a novel test point, we estimate both the composition and size of the training subset likely to yield an accurate local model. We demonstrate the approach on image classification tasks on SUN and aPascal and show it outperforms traditional global and local approaches."

We study the sensitivity of a MAP configuration of a discrete probabilistic graphical model with respect to perturbations of its parameters. These perturbations are global, in the sense that simultaneous perturbations of all the parameters (or any chosen subset of them) are allowed. Our main contribution is an exact algorithm that can check whether the MAP configuration is robust with respect to given perturbations. Its complexity is essentially the same as that of obtaining the MAP configuration itself, so it can be promptly used with minimal effort. We use our algorithm to identify the largest global perturbation that does not induce a change in the MAP configuration, and we successfully apply this robustness measure in two practical scenarios: the prediction of facial action units with posed images and the classification of multiple real public data sets. A strong correlation between the proposed robustness measure and accuracy is verified in both scenarios.

High-dimensional, simultaneous recordings of neural spiking activity are often explored, analyzed and visualized with the help of latent variable or factor models. Such models are however ill-equipped to extract structure beyond shared, distributed aspects of firing activity across multiple cells. Here, we extend unstructured factor models by proposing a model that discovers subpopulations or groups of cells from the pool of recorded neurons. The model combines aspects of mixture of factor analyzer models for capturing clustering structure, and aspects of latent dynamical system models for capturing temporal dependencies. In the resulting model, we infer the subpopulations and the latent factors from data using variational inference and model parameters are estimated by Expectation Maximization (EM). We also address the crucial problem of initializing parameters for EM by extending a sparse subspace clustering algorithm to integer-valued spike count observations. We illustrate the merits of the proposed model by applying it to calcium-imaging data from spinal cord neurons, and we show that it uncovers meaningful clustering structure in the data.

We propose a pre-training technique for recurrent neural networks based on linear autoencoder networks for sequences, i.e. linear dynamical systems modelling the target sequences. We start by giving a closed form solution for the definition of the optimal weights of a linear autoencoder given a training set of sequences. This solution, however, is computationally very demanding, so we suggest a procedure to get an approximate solution for a given number of hidden units. The weights obtained for the linear autoencoder are then used as initial weights for the input-to-hidden connections of a recurrent neural network, which is then trained on the desired task. Using four well known datasets of sequences of polyphonic music, we show that the proposed pre-training approach is highly effective, since it allows to largely improve the state of the art results on all the considered datasets.

Training of the neural autoregressive density estimator (NADE) can be viewed as doing one step of probabilistic inference on missing values in data. We propose a new model that extends this inference scheme to multiple steps, arguing that it is easier to learn to improve a reconstruction in $k$ steps rather than to learn to reconstruct in a single inference step. The proposed model is an unsupervised building block for deep learning that combines the desirable properties of NADE and multi-predictive training: (1) Its test likelihood can be computed analytically, (2) it is easy to generate independent samples from it, and (3) it uses an inference engine that is a superset of variational inference for Boltzmann machines. The proposed NADE-k is competitive with the state-of-the-art in density estimation on the two datasets tested.