We propose efficient algorithms for simultaneous clustering and completion of incomplete high-dimensional data that lie in a union of low-dimensional subspaces. We cast the problem as finding a completion of the data matrix so that each point can be reconstructed as a linear or affine combination of a few data points. Since the problem is NP-hard, we propose a lifting framework and reformulate the problem as a group-sparse recovery of each incomplete data point in a dictionary built using incomplete data, subject to rank-one constraints. To solve the problem efficiently, we propose a rank pursuit algorithm and a convex relaxation. The solution of our algorithms recover missing entries and provides a similarity matrix for clustering. Our algorithms can deal with both low-rank and high-rank matrices, does not suffer from initialization, does not need to know dimensions of subspaces and can work with a small number of data points. By extensive experiments on synthetic data and real problems of video motion segmentation and completion of motion capture data, we show that when the data matrix is low-rank, our algorithm performs on par with or better than low-rank matrix completion methods, while for high-rank data matrices, our method significantly outperforms existing algorithms.

Cross-region dynamic connectivity, which describes the spatio-temporal dependence of neural activity among multiple brain regions of interest (ROIs), can provide important information for understanding cognition. For estimating such connectivity, magnetoencephalography (MEG) and electroencephalography (EEG) are well-suited tools because of their millisecond temporal resolution. However, localizing source activity in the brain requires solving an under-determined linear problem. In typical two-step approaches, researchers first solve the linear problem with generic priors assuming independence across ROIs, and secondly quantify cross-region connectivity. In this work, we propose a one-step state-space model to improve estimation of dynamic connectivity. The model treats the mean activity in individual ROIs as the state variable and describes non-stationary dynamic dependence across ROIs using time-varying auto-regression. Compared with a two-step method, which first obtains the commonly used minimum-norm estimates of source activity, and then fits the auto-regressive model, our state-space model yielded smaller estimation errors on simulated data where the model assumptions held. When applied on empirical MEG data from one participant in a scene-processing experiment, our state-space model also demonstrated intriguing preliminary results, indicating leading and lagged linear dependence between the early visual cortex and a higher-level scene-sensitive region, which could reflect feedforward and feedback information flow within the visual cortex during scene processing.

We present a general framework for classification of sparse and irregularly-sampled time series. The properties of such time series can result in substantial uncertainty about the values of the underlying temporal processes, while making the data difficult to deal with using standard classification methods that assume fixeddimensional feature spaces. To address these challenges, we propose an uncertaintyaware classification framework based on a special computational layer we refer to as the Gaussian process adapter that can connect irregularly sampled time series data to any black-box classifier learnable using gradient descent. We show how to scale up the required computations based on combining the structured kernel interpolation framework and the Lanczos approximation method, and how to discriminatively train the Gaussian process adapter in combination with a number of classifiers end-to-end using backpropagation.

Stochastic gradient-based Monte Carlo methods such as stochastic gradient Langevin dynamics are useful tools for posterior inference on large scale datasets in many machine learning applications. These methods scale to large datasets by using noisy gradients calculated using a mini-batch or subset of the dataset. However, the high variance inherent in these noisy gradients degrades performance and leads to slower mixing. In this paper, we present techniques for reducing variance in stochastic gradient Langevin dynamics, yielding novel stochastic Monte Carlo methods that improve performance by reducing the variance in the stochastic gradient. We show that our proposed method has better theoretical guarantees on convergence rate than stochastic Langevin dynamics. This is complemented by impressive empirical results obtained on a variety of real world datasets, and on four different machine learning tasks (regression, classification, independent component analysis and mixture modeling). These theoretical and empirical contributions combine to make a compelling case for using variance reduction in stochastic Monte Carlo methods.

Tensor factorization is a powerful tool to analyse multi-way data. Recently proposed nonlinear factorization methods, although capable of capturing complex relationships, are computationally quite expensive and may suffer a severe learning bias in case of extreme data sparsity. Therefore, we propose a distributed, flexible nonlinear tensor factorization model, which avoids the expensive computations and structural restrictions of the Kronecker-product in the existing TGP formulations, allowing an arbitrary subset of tensorial entries to be selected for training. Meanwhile, we derive a tractable and tight variational evidence lower bound (ELBO) that enables highly decoupled, parallel computations and high-quality inference.

We propose a novel extension of the encoder-decoder framework, called a review network. The review network is generic and can enhance any existing encoderdecoder model: in this paper, we consider RNN decoders with both CNN and RNN encoders. The review network performs a number of review steps with attention mechanism on the encoder hidden states, and outputs a thought vector after each review step; the thought vectors are used as the input of the attention mechanism in the decoder. We show that conventional encoder-decoders are a special case of our framework. Empirically, we show that our framework improves over state-ofthe-art encoder-decoder systems on the tasks of image captioning and source code captioning.

We study optimization of finite sums of geodesically smooth functions on Riemannian manifolds. Although variance reduction techniques for optimizing finite-sums have witnessed tremendous attention in the recent years, existing work is limited to vector space problems.

We introduce the Adaptive Skills, Adaptive Partitions (ASAP) framework that (1) learns skills (i.e., temporally extended actions or options) as well as (2) where to apply them. We believe that both (1) and (2) are necessary for a truly general skill learning framework, which is a key building block needed to scale up to lifelong learning agents. The ASAP framework can also solve related new tasks simply by adapting where it applies its existing learned skills. We prove that ASAP converges to a local optimum under natural conditions. Finally, our experimental results, which include a RoboCup domain, demonstrate the ability of ASAP to learn where to reuse skills as well as solve multiple tasks with considerably less experience than solving each task from scratch.

Non-negative matrix factorization is a popular tool for decomposing data into feature and weight matrices under non-negativity constraints. It enjoys practical success but is poorly understood theoretically. This paper proposes an algorithm that alternates between decoding the weights and updating the features, and shows that assuming a generative model of the data, it provably recovers the groundtruth under fairly mild conditions. In particular, its only essential requirement on features is linear independence. Furthermore, the algorithm uses ReLU to exploit the non-negativity for decoding the weights, and thus can tolerate adversarial noise that can potentially be as large as the signal, and can tolerate unbiased noise much larger than the signal. The analysis relies on a carefully designed coupling between two potential functions, which we believe is of independent interest.

We propose a framework for Semi-Supervised Active Clustering framework (SSAC), where the learner is allowed to interact with a domain expert, asking whether two given instances belong to the same cluster or not. We study the query and computational complexity of clustering in this framework. We consider a setting where the expert conforms to a center-based clustering with a notion of margin. We show that there is a trade off between computational complexity and query complexity; We prove that for the case of k-means clustering (i.e., when the expert conforms to a solution of k-means), having access to relatively few such queries allows efficient solutions to otherwise NP hard problems.