Ali Siahkamari

Ali Siahkamari, Durmus Alp Emre Acar, Christopher Liao et al.

The task of approximating an arbitrary convex function arises in several learning problems such as convex regression, learning with a difference of convex (DC) functions, and learning Bregman or $f$-divergences. In this paper, we develop and analyze an approach for solving a broad range of convex function learning problems that is faster than state-of-the-art approaches. Our approach is based on a 2-block ADMM method where each block can be computed in closed form. For the task of convex Lipschitz regression, we establish that our proposed algorithm converges with iteration complexity of $ O(n\sqrt{d}/ε)$ for a dataset $\bm X \in \mathbb R^{n\times d}$ and $ε> 0$. Combined with per-iteration computation complexity, our method converges with the rate $O(n^3 d^{1.5}/ε+n^2 d^{2.5}/ε+n d^3/ε)$. This new rate improves the state of the art rate of $O(n^5d^2/ε)$ if $d = o( n^4)$. Further we provide similar solvers for DC regression and Bregman divergence learning. Unlike previous approaches, our method is amenable to the use of GPUs. We demonstrate on regression and metric learning experiments that our approach is over 100 times faster than existing approaches on some data sets, and produces results that are comparable to state of the art.

Ali Siahkamari, Aditya Gangrade, Brian Kulis et al.

We present a new piecewise linear regression methodology that utilizes fitting a difference of convex functions (DC functions) to the data. These are functions $f$ that may be represented as the difference $φ_1 - φ_2$ for a choice of convex functions $φ_1, φ_2$. The method proceeds by estimating piecewise-liner convex functions, in a manner similar to max-affine regression, whose difference approximates the data. The choice of the function is regularised by a new seminorm over the class of DC functions that controls the $\ell_\infty$ Lipschitz constant of the estimate. The resulting methodology can be efficiently implemented via Quadratic programming even in high dimensions, and is shown to have close to minimax statistical risk. We empirically validate the method, showing it to be practically implementable, and to have comparable performance to existing regression/classification methods on real-world datasets.

Ali Siahkamari, Xide Xia, Venkatesh Saligrama et al.

Bregman divergences generalize measures such as the squared Euclidean distance and the KL divergence, and arise throughout many areas of machine learning. In this paper, we focus on the problem of approximating an arbitrary Bregman divergence from supervision, and we provide a well-principled approach to analyzing such approximations. We develop a formulation and algorithm for learning arbitrary Bregman divergences based on approximating their underlying convex generating function via a piecewise linear function. We provide theoretical approximation bounds using our parameterization and show that the generalization error $O_p(m^{-1/2})$ for metric learning using our framework matches the known generalization error in the strictly less general Mahalanobis metric learning setting. We further demonstrate empirically that our method performs well in comparison to existing metric learning methods, particularly for clustering and ranking problems.

Rachel Manzelli, Vijay Thakkar, Ali Siahkamari et al.

Existing automatic music generation approaches that feature deep learning can be broadly classified into two types: raw audio models and symbolic models. Symbolic models, which train and generate at the note level, are currently the more prevalent approach; these models can capture long-range dependencies of melodic structure, but fail to grasp the nuances and richness of raw audio generations. Raw audio models, such as DeepMind's WaveNet, train directly on sampled audio waveforms, allowing them to produce realistic-sounding, albeit unstructured music. In this paper, we propose an automatic music generation methodology combining both of these approaches to create structured, realistic-sounding compositions. We consider a Long Short Term Memory network to learn the melodic structure of different styles of music, and then use the unique symbolic generations from this model as a conditioning input to a WaveNet-based raw audio generator, creating a model for automatic, novel music. We then evaluate this approach by showcasing results of this work.