We present Neumann eigenmaps (NeuMaps), a novel approach for enhancing the standard diffusion map embedding using landmarks, i.e distinguished samples within the dataset. By interpreting these landmarks as a subgraph of the larger data graph, NeuMaps are obtained via the eigendecomposition of a renormalized Neumann Laplacian. We show that NeuMaps offer two key advantages: (1) they provide a computationally efficient embedding that accurately recovers the diffusion distance associated with the reflecting random walk on the subgraph, and (2) they naturally incorporate the Nystr\"om extension within the diffusion map framework through the discrete Neumann boundary condition. Through examples in digit classification and molecular dynamics, we demonstrate that NeuMaps not only improve upon existing landmark-based embedding methods but also enhance the stability of diffusion map embeddings to the removal of highly significant points.

In this paper, we study the estimation of the k-dimensional sparse principal subspace of covariance matrix Σ in the high-dimensional setting. We aim to recover the oracle principal subspace solution, i.e., the principal subspace estimator obtained assuming the true support is known a priori. To this end, we propose a family of estimators based on the semidefinite relaxation of sparse PCA with novel regularizations. In particular, under a weak assumption on the magnitude of the population projection matrix, one estimator within this family exactly recovers the true support with high probability, has exact rank-k, and attains a s/n statistical rate of convergence with s being the subspace sparsity level and n the sample size. Compared to existing support recovery results for sparse PCA, our approach does not hinge on the spiked covariance model or the limited correlation condition. As a complement to the first estimator that enjoys the oracle property, we prove that, another estimator within the family achieves a sharper statistical rate of convergence than the standard semidefinite relaxation of sparse PCA, even when the previous assumption on the magnitude of the projection matrix is violated.

In this work we introduce a new optimisation method called SAGA in the spirit of SAG, SDCA, MISO and SVRG, a set of recently proposed incremental gradient algorithms with fast linear convergence rates. SAGA improves on the theory behind SAG and SVRG, with better theoretical convergence rates, and has support for composite objectives where a proximal operator is used on the regulariser. Unlike SDCA, SAGA supports non-strongly convex problems directly, and is adaptive to any inherent strong convexity of the problem. We give experimental results showing the effectiveness of our method.

To alleviate this deficiency, we propose three complementary approaches.

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.

In sparse principal component analysis we are given noisy observations of a lowrank matrix of dimension n p and seek to reconstruct it under additional sparsity assumptions.

L-BFGS has been applied as an effective parameter estimation method for various machine learning algorithms since 1980s. With an increasing demand to deal with massive instances and variables, it is important to scale up and parallelize L-BFGS effectively in a distributed system. In this paper, we study the problem of parallelizing the L-BFGS algorithm in large clusters of tens of thousands of shared-nothing commodity machines. First, we show that a naive implementation of L-BFGS using Map-Reduce requires either a significant amount of memory or a large number of map-reduce steps with negative performance impact. Second, we propose a new L-BFGS algorithm, called Vector-free L-BFGS, which avoids the expensive dot product operations in the two loop recursion and greatly improves computation efficiency with a great degree of parallelism. The algorithm scales very well and enables a variety of machine learning algorithms to handle a massive number of variables over large datasets. We prove the mathematical equivalence of the new Vector-free L-BFGS and demonstrate its excellent performance and scalability using real-world machine learning problems with billions of variables in production clusters.

We investigate a general framework of multiplicative multitask feature learning which decomposes each task's model parameters into a multiplication of two components. One of the components is used across all tasks and the other component is task-specific. Several previous methods have been proposed as special cases of our framework. We study the theoretical properties of this framework when different regularization conditions are applied to the two decomposed components. We prove that this framework is mathematically equivalent to the widely used multitask feature learning methods that are based on a joint regularization of all model parameters, but with a more general form of regularizers. Further, an analytical formula is derived for the across-task component as related to the taskspecific component for all these regularizers, leading to a better understanding of the shrinkage effect.

In this paper we propose a general framework for learning distributed representations of attributes: characteristics of text whose representations can be jointly learned with word embeddings. Attributes can correspond to a wide variety of concepts, such as document indicators (to learn sentence vectors), language indicators (to learn distributed language representations), meta-data and side information (such as the age, gender and industry of a blogger) or representations of authors. We describe a third-order model where word context and attribute vectors interact multiplicatively to predict the next word in a sequence. This leads to the notion of conditional word similarity: how meanings of words change when conditioned on different attributes. We perform several experimental tasks including sentiment classification, cross-lingual document classification, and blog authorship attribution. We also qualitatively evaluate conditional word neighbours and attribute-conditioned text generation.

Stochastic variational inference (SVI) lets us scale up Bayesian computation to massive data. It uses stochastic optimization to fit a variational distribution, following easy-to-compute noisy natural gradients. As with most traditional stochastic optimization methods, SVI takes precautions to use unbiased stochastic gradients whose expectations are equal to the true gradients. In this paper, we explore the idea of following biased stochastic gradients in SVI. Our method replaces the natural gradient with a similarly constructed vector that uses a fixed-window moving average of some of its previous terms. We will demonstrate the many advantages of this technique. First, its computational cost is the same as for SVI and storage requirements only multiply by a constant factor. Second, it enjoys significant variance reduction over the unbiased estimates, smaller bias than averaged gradients, and leads to smaller mean-squared error against the full gradient. We test our method on latent Dirichlet allocation with three large corpora.