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Shape constraints such as positive semi-definiteness (PSD) for matrices or convexity for functions play a central role in many applications in machine learning and sciences, including metric learning, optimal transport, and economics. Yet, very few function models exist that enforce PSD-ness or convexity with good empirical performance and theoretical guarantees. In this paper, we introduce a kernel sum-of-squares model for functions that take values in the PSD cone, which extends kernel sums-of-squares models that were recently proposed to encode non-negative scalar functions. We provide a representer theorem for this class of PSD functions, show that it constitutes a universal approximator of PSD functions, and derive eigenvalue bounds in the case of subsampled equality constraints. We then apply our results to modeling convex functions, by enforcing a kernel sum-of-squares representation of their Hessian, and show that any smooth and strongly convex function may be thus represented. Finally, we illustrate our methods on a PSD matrix-valued regression task, and on scalar-valued convex regression.

The discovery of drug-target interactions (DTIs) is a very promising area of research with great potential. In general, the identification of reliable interactions among drugs and proteins can boost the development of effective pharmaceuticals. In this work, we leverage random walks and matrix factorization techniques towards DTI prediction. In particular, we take a multi-layered network perspective, where different layers correspond to different similarity metrics between drugs and targets. To fully take advantage of topology information captured in multiple views, we develop an optimization framework, called MDMF, for DTI prediction. The framework learns vector representations of drugs and targets that not only retain higher-order proximity across all hyper-layers and layer-specific local invariance, but also approximates the interactions with their inner product. Furthermore, we propose an ensemble method, called MDMF2A, which integrates two instantiations of the MDMF model that optimize surrogate losses of the area under the precision-recall curve (AUPR) and the area under the receiver operating characteristic curve (AUC), respectively. The empirical study on real-world DTI datasets shows that our method achieves significant improvement over current state-of-the-art approaches in four different settings. Moreover, the validation of highly ranked non-interacting pairs also demonstrates the potential of MDMF2A to discover novel DTIs.

The generalization capacity of various machine learning models exhibits different phenomena in the under- and over-parameterized regimes. In this paper, we focus on regression models such as feature regression and kernel regression and analyze a generalized weighted least-squares optimization method for computational learning and inversion with noisy data. The highlight of the proposed framework is that we allow weighting in both the parameter space and the data space. The weighting scheme encodes both a priori knowledge on the object to be learned and a strategy to weight the contribution of different data points in the loss function. Here, we characterize the impact of the weighting scheme on the generalization error of the learning method, where we derive explicit generalization errors for the random Fourier feature model in both the under- and over-parameterized regimes. For more general feature maps, error bounds are provided based on the singular values of the feature matrix. We demonstrate that appropriate weighting from prior knowledge can improve the generalization capability of the learned model.

We propose a novel multibody dynamics simulation framework that can efficiently deal with large-dimensionality and complementarity multi-contact conditions. Typical contact simulation approaches perform contact impulse-level fixed-point iteration (IL-FPI), which has high time-complexity from large-size matrix inversion and multiplication, as well as susceptibility to ill-conditioned contact situations. To circumvent this, we propose a novel framework based on velocity-level fixed-point iteration (VL-FPI), which, by utilizing a certain surrogate dynamics and contact nodalization (with virtual nodes), can achieve not only inter-contact decoupling but also their inter-axes decoupling (i.e., contact diagonalization). This then enables us to one-shot/parallel-solve the contact problem during each VL-FPI iteration-loop, while the surrogate dynamics structure allows us to circumvent large-size/dense matrix inversion/multiplication, thereby, significantly speeding up the simulation time with improved convergence property. We theoretically show that the solution of our framework is consistent with that of the original problem and, further, elucidate mathematical conditions for the convergence of our proposed solver. Performance and properties of our proposed simulation framework are also demonstrated and experimentally-validated for various large-dimensional/multi-contact scenarios including deformable objects.

Given input-output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically-rigorous scheme for learning the associated Green's function $G$. By exploiting the hierarchical low-rank structure of $G$, we show that one can construct an approximant to $G$ that converges almost surely and achieves a relative error of $\mathcal{O}(\Gamma_\epsilon^{-1/2}\log^3(1/\epsilon)\epsilon)$ using at most $\mathcal{O}(\epsilon^{-6}\log^4(1/\epsilon))$ input-output training pairs with high probability, for any $0<\epsilon<1$. The quantity $0<\Gamma_\epsilon\leq 1$ characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert--Schmidt operators and characterize the quality of covariance kernels for PDE learning.

In health-pollution cohort studies, accurate predictions of pollutant concentrations at new locations are needed, since the locations of fixed monitoring sites and study participants are often spatially misaligned. For multi-pollution data, principal component analysis (PCA) is often incorporated to obtain low-rank (LR) structure of the data prior to spatial prediction. Recently developed predictive PCA modifies the traditional algorithm to improve the overall predictive performance by leveraging both LR and spatial structures within the data. However, predictive PCA requires complete data or an initial imputation step. Nonparametric imputation techniques without accounting for spatial information may distort the underlying structure of the data, and thus further reduce the predictive performance. We propose a convex optimization problem inspired by the LR matrix completion framework and develop a proximal algorithm to solve it. Missing data are imputed and handled concurrently within the algorithm, which eliminates the necessity of a separate imputation step. We show that our algorithm has low computational burden and leads to reliable predictive performance as the severity of missing data increases.

Because of the superior feature representation ability of deep learning, various deep Click-Through Rate (CTR) models are deployed in the commercial systems by industrial companies. To achieve better performance, it is necessary to train the deep CTR models on huge volume of training data efficiently, which makes speeding up the training process an essential problem. Different from the models with dense training data, the training data for CTR models is usually high-dimensional and sparse. To transform the high-dimensional sparse input into low-dimensional dense real-value vectors, almost all deep CTR models adopt the embedding layer, which easily reaches hundreds of GB or even TB. Since a single GPU cannot afford to accommodate all the embedding parameters, when performing distributed training, it is not reasonable to conduct the data-parallelism only. Therefore, existing distributed training platforms for recommendation adopt model-parallelism. Specifically, they use CPU (Host) memory of servers to maintain and update the embedding parameters and utilize GPU worker to conduct forward and backward computations. Unfortunately, these platforms suffer from two bottlenecks: (1) the latency of pull \& push operations between Host and GPU; (2) parameters update and synchronization in the CPU servers. To address such bottlenecks, in this paper, we propose the ScaleFreeCTR: a MixCache-based distributed training system for CTR models. Specifically, in SFCTR, we also store huge embedding table in CPU but utilize GPU instead of CPU to conduct embedding synchronization efficiently. To reduce the latency of data transfer between both GPU-Host and GPU-GPU, the MixCache mechanism and Virtual Sparse Id operation are proposed. Comprehensive experiments and ablation studies are conducted to demonstrate the effectiveness and efficiency of SFCTR.

Several recent applications of optimal transport (OT) theory to machine learning have relied on regularization, notably entropy and the Sinkhorn algorithm. Because matrix-vector products are pervasive in the Sinkhorn algorithm, several works have proposed to \textit{approximate} kernel matrices appearing in its iterations using low-rank factors. Another route lies instead in imposing low-rank constraints on the feasible set of couplings considered in OT problems, with no approximations on cost nor kernel matrices. This route was first explored by Forrow et al., 2018, who proposed an algorithm tailored for the squared Euclidean ground cost, using a proxy objective that can be solved through the machinery of regularized 2-Wasserstein barycenters. Building on this, we introduce in this work a generic approach that aims at solving, in full generality, the OT problem under low-rank constraints with arbitrary costs. Our algorithm relies on an explicit factorization of low rank couplings as a product of \textit{sub-coupling} factors linked by a common marginal; similar to an NMF approach, we alternatively updates these factors. We prove the non-asymptotic stationary convergence of this algorithm and illustrate its efficiency on benchmark experiments.

Tracking by detection is a common approach to solving the Multiple Object Tracking problem. In this paper we show how deep metric learning can be used to improve three aspects of tracking by detection. We train a convolutional neural network to learn an embedding function in a Siamese configuration on a large person re-identification dataset offline. It is then used to improve the online performance of tracking while retaining a high frame rate. We use this learned appearance metric to robustly build estimates of pedestrian's trajectories in the MOT16 dataset. In breaking with the tracking by detection model, we use our appearance metric to propose detections using the predicted state of a tracklet as a prior in the case where the detector fails. This method achieves competitive results in evaluation, especially among online, real-time approaches. We present an ablative study showing the impact of each of the three uses of our deep appearance metric.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.