Observed Universality of Phase Transitions in High-Dimensional Geometry, with Implications for Modern Data Analysis and Signal Processing

We review connections between phase transitions in high-dimensional combinatorial geometry and phase transitions occurring in modern high-dimensional data analysis and signal processing. In data analysis, such transitions arise as abrupt breakdown of linear model selection, robust data fitting or compressed sensing reconstructions, when the complexity of the model or the number of outliers increases beyond a threshold. In combinatorial geometry these transitions appear as abrupt changes in the properties of face counts of convex polytopes when the dimensions are varied. The thresholds in these very different problems appear in the same critical locations after appropriate calibration of variables. These thresholds are important in each subject area: for linear modelling, they place hard limits on the degree to which the now-ubiquitous high-throughput data analysis can be successful; for robustness, they place hard limits on the degree to which standard robust fitting methods can tolerate outliers before breaking down; for compressed sensing, they define the sharp boundary of the undersampling/sparsity tradeoff in undersampling theorems. Existing derivations of phase transitions in combinatorial geometry assume the underlying matrices have independent and identically distributed (iid) Gaussian elements. In applications, however, it often seems that Gaussianity is not required. We conducted an extensive computational experiment and formal inferential analysis to test the hypothesis that these phase transitions are {\it universal} across a range of underlying matrix ensembles. The experimental results are consistent with an asymptotic large-$n$ universality across matrix ensembles; finite-sample universality can be rejected.Comment: 47 pages, 24 figures, 10 table

Fractals in the Nervous System: conceptual Implications for Theoretical Neuroscience

This essay is presented with two principal objectives in mind: first, to document the prevalence of fractals at all levels of the nervous system, giving credence to the notion of their functional relevance; and second, to draw attention to the as yet still unresolved issues of the detailed relationships among power law scaling, self-similarity, and self-organized criticality. As regards criticality, I will document that it has become a pivotal reference point in Neurodynamics. Furthermore, I will emphasize the not yet fully appreciated significance of allometric control processes. For dynamic fractals, I will assemble reasons for attributing to them the capacity to adapt task execution to contextual changes across a range of scales. The final Section consists of general reflections on the implications of the reviewed data, and identifies what appear to be issues of fundamental importance for future research in the rapidly evolving topic of this review

The generalized Lasso with non-linear observations

We study the problem of signal estimation from non-linear observations when the signal belongs to a low-dimensional set buried in a high-dimensional space. A rough heuristic often used in practice postulates that non-linear observations may be treated as noisy linear observations, and thus the signal may be estimated using the generalized Lasso. This is appealing because of the abundance of efficient, specialized solvers for this program. Just as noise may be diminished by projecting onto the lower dimensional space, the error from modeling non-linear observations with linear observations will be greatly reduced when using the signal structure in the reconstruction. We allow general signal structure, only assuming that the signal belongs to some set K in R^n. We consider the single-index model of non-linearity. Our theory allows the non-linearity to be discontinuous, not one-to-one and even unknown. We assume a random Gaussian model for the measurement matrix, but allow the rows to have an unknown covariance matrix. As special cases of our results, we recover near-optimal theory for noisy linear observations, and also give the first theoretical accuracy guarantee for 1-bit compressed sensing with unknown covariance matrix of the measurement vectors.Comment: 21 page

Determination of Nonlinear Genetic Architecture using Compressed Sensing

We introduce a statistical method that can reconstruct nonlinear genetic models (i.e., including epistasis, or gene-gene interactions) from phenotype-genotype (GWAS) data. The computational and data resource requirements are similar to those necessary for reconstruction of linear genetic models (or identification of gene-trait associations), assuming a condition of generalized sparsity, which limits the total number of gene-gene interactions. An example of a sparse nonlinear model is one in which a typical locus interacts with several or even many others, but only a small subset of all possible interactions exist. It seems plausible that most genetic architectures fall in this category. Our method uses a generalization of compressed sensing (L1-penalized regression) applied to nonlinear functions of the sensing matrix. We give theoretical arguments suggesting that the method is nearly optimal in performance, and demonstrate its effectiveness on broad classes of nonlinear genetic models using both real and simulated human genomes.Comment: 20 pages, 8 figures. arXiv admin note: text overlap with arXiv:1408.342

Convexity in source separation: Models, geometry, and algorithms

Source separation or demixing is the process of extracting multiple components entangled within a signal. Contemporary signal processing presents a host of difficult source separation problems, from interference cancellation to background subtraction, blind deconvolution, and even dictionary learning. Despite the recent progress in each of these applications, advances in high-throughput sensor technology place demixing algorithms under pressure to accommodate extremely high-dimensional signals, separate an ever larger number of sources, and cope with more sophisticated signal and mixing models. These difficulties are exacerbated by the need for real-time action in automated decision-making systems. Recent advances in convex optimization provide a simple framework for efficiently solving numerous difficult demixing problems. This article provides an overview of the emerging field, explains the theory that governs the underlying procedures, and surveys algorithms that solve them efficiently. We aim to equip practitioners with a toolkit for constructing their own demixing algorithms that work, as well as concrete intuition for why they work