Quantized Compressive K-Means

The recent framework of compressive statistical learning aims at designing tractable learning algorithms that use only a heavily compressed representation-or sketch-of massive datasets. Compressive K-Means (CKM) is such a method: it estimates the centroids of data clusters from pooled, non-linear, random signatures of the learning examples. While this approach significantly reduces computational time on very large datasets, its digital implementation wastes acquisition resources because the learning examples are compressed only after the sensing stage. The present work generalizes the sketching procedure initially defined in Compressive K-Means to a large class of periodic nonlinearities including hardware-friendly implementations that compressively acquire entire datasets. This idea is exemplified in a Quantized Compressive K-Means procedure, a variant of CKM that leverages 1-bit universal quantization (i.e. retaining the least significant bit of a standard uniform quantizer) as the periodic sketch nonlinearity. Trading for this resource-efficient signature (standard in most acquisition schemes) has almost no impact on the clustering performances, as illustrated by numerical experiments

Spatial Random Sampling: A Structure-Preserving Data Sketching Tool

Random column sampling is not guaranteed to yield data sketches that preserve the underlying structures of the data and may not sample sufficiently from less-populated data clusters. Also, adaptive sampling can often provide accurate low rank approximations, yet may fall short of producing descriptive data sketches, especially when the cluster centers are linearly dependent. Motivated by that, this paper introduces a novel randomized column sampling tool dubbed Spatial Random Sampling (SRS), in which data points are sampled based on their proximity to randomly sampled points on the unit sphere. The most compelling feature of SRS is that the corresponding probability of sampling from a given data cluster is proportional to the surface area the cluster occupies on the unit sphere, independently from the size of the cluster population. Although it is fully randomized, SRS is shown to provide descriptive and balanced data representations. The proposed idea addresses a pressing need in data science and holds potential to inspire many novel approaches for analysis of big data

Scalable and Robust Community Detection with Randomized Sketching

This paper explores and analyzes the unsupervised clustering of large partially observed graphs. We propose a scalable and provable randomized framework for clustering graphs generated from the stochastic block model. The clustering is first applied to a sub-matrix of the graph's adjacency matrix associated with a reduced graph sketch constructed using random sampling. Then, the clusters of the full graph are inferred based on the clusters extracted from the sketch using a correlation-based retrieval step. Uniform random node sampling is shown to improve the computational complexity over clustering of the full graph when the cluster sizes are balanced. A new random degree-based node sampling algorithm is presented which significantly improves upon the performance of the clustering algorithm even when clusters are unbalanced. This algorithm improves the phase transitions for matrix-decomposition-based clustering with regard to computational complexity and minimum cluster size, which are shown to be nearly dimension-free in the low inter-cluster connectivity regime. A third sampling technique is shown to improve balance by randomly sampling nodes based on spatial distribution. We provide analysis and numerical results using a convex clustering algorithm based on matrix completion