Sketching for large-scale learning of mixture models

Abstract

Learning parameters from voluminous data can be prohibitive in terms of memory and computational requirements. We propose a "compressive learning" framework where we estimate model parameters from a sketch of the training data. This sketch is a collection of generalized moments of the underlying probability distribution of the data. It can be computed in a single pass on the training set, and is easily computable on streams or distributed datasets. The proposed framework shares similarities with compressive sensing, which aims at drastically reducing the dimension of high-dimensional signals while preserving the ability to reconstruct them.To perform the estimation task, we derive an iterative algorithm analogous to sparse reconstruction algorithms in the context of linear inverse problems. We exemplify our framework with the compressive estimation of a Gaussian Mixture Model (GMM), providing heuristics on the choice of the sketching procedure and theoretical guarantees of reconstruction. We experimentally show on synthetic data that the proposed algorithm yields results comparable to the classical ExpectationMaximization (EM) technique while requiring significantly less memory and fewer computations when the number of database elements is large. We further demonstrate the potential of the approach on real large-scale data (over 10 8 training samples) for the task of model-based speaker verification. Finally, we draw some connections between the proposed framework and approximate Hilbert space embedding of probability distributions using random features. We show that the proposed sketching operator can be seen as an innovative method to design translation-invariant kernels adapted to the analysis of GMMs. We also use this theoretical framework to derive information preservation guarantees, in the spirit of infinite-dimensional compressive sensing.

Compressive K-means

“…Some techniques might further reduce these complexities. As detailed in [23], most operations in CKM can be narrowed down to performing multiplications by W and W T . Therefore, both computing the sketch and performing CKM could benefit from the replacement of W by a suitably randomized fast transform [6,7].…”

“…We compare our Matlab implementation of CKM, available at [23], with Matlab's kmeans function that implements Lloyd-Max.…”

Compressive statistical learning with random feature moments

“…-For Compressive k-means and Compressive Gaussian Mixture Modeling (cf the companion paper [Gribonval et al, 2020]), the resulting optimization problem has been empirically addressed through the CL-OMPR algorithm [Keriven et al, 2015[Keriven et al, , 2016. Algorithmic success guarantees are an interesting challenge.…”

Sketched Clustering via Hybrid Approximate Message Passing

“…In this section, we present the results of two numerical experiments used to test the performance of the CL-AMP, CL-OMPR, and k-means++ algorithms. For k-means++, we used the implementation provided by MATLAB and, for CL-OMPR, we downloaded the MATLAB implementation from [17] and enabled the "++" initialization method. CL-OMPR and CL-AMP used the same sketch y, whose frequency vectors W were drawn using the method described in [5].…”