Romeil Sandhu

Graph Curvature for Differentiating Cancer Networks

Cellular interactions can be modeled as complex dynamical systems represented by weighted graphs. The functionality of such networks, including measures of robustness, reliability, performance, and efficiency, are intrinsically tied to the topology and geometry of the underlying graph. Utilizing recently proposed geometric notions of curvature on weighted graphs, we investigate the features of gene co-expression networks derived from large-scale genomic studies of cancer. We find that the curvature of these networks reliably distinguishes between cancer and normal samples, with cancer networks exhibiting higher curvature than their normal counterparts. We establish a quantitative relationship between our findings and prior investigations of network entropy. Furthermore, we demonstrate how our approach yields additional, non-trivial pair-wise (i.e. gene-gene) interactions which may be disrupted in cancer samples. The mathematical formulation of our approach yields an exact solution to calculating pair-wise changes in curvature which was computationally infeasible using prior methods. As such, our findings lay the foundation for an analytical approach to studying complex biological networks.

Ricci curvature: An economic indicator for market fragility and systemic risk.

Researchers introduce the geometric concept of curvature toward the application of uncovering systemic risk and market fragility. Quantifying the systemic risk and fragility of financial systems is of vital importance in analyzing market efficiency, deciding on portfolio allocation, and containing financial contagions. At a high level, financial systems may be represented as weighted graphs that characterize the complex web of interacting agents and information flow (for example, debt, stock returns, and shareholder ownership). Such a representation often turns out to provide keen insights. We show that fragility is a system-level characteristic of “business-as-usual” market behavior and that financial crashes are invariably preceded by system-level changes in robustness. This was done by leveraging previous work, which suggests that Ricci curvature, a key geometric feature of a given network, is negatively correlated to increases in network fragility. To illustrate this insight, we examine daily returns from a set of stocks comprising the Standard and Poor’s 500 (S&P 500) over a 15-year span to highlight the fact that corresponding changes in Ricci curvature constitute a financial “crash hallmark.” This work lays the foundation of understanding how to design (banking) systems and policy regulations in a manner that can combat financial instabilities exposed during the 2007–2008 crisis.

Pediatric Sarcoma Data Forms a Unique Cluster Measured via the Earth Mover’s Distance

In this note, we combined pediatric sarcoma data from Columbia University with adult sarcoma data collected from TCGA, in order to see if one can automatically discern a unique pediatric cluster in the combined data set. Using a novel clustering pipeline based on optimal transport theory, this turned out to be the case. The overall methodology may find uses for the classification of data from other biological networking problems.

Machine Learning for Joint Classification and Segmentation

In this note, we consider the use of 3D models for visual tracking in controlled active vision. The models are used for a joint 2D segmentation/3D pose estimation procedure in which we automatically couple the two processes under one energy functional. Further, employing principal component analysis or locally linear embedding from statistical learning, one can train our tracker on a catalog of 3D shapes, giving a priori shape information. The segmentation itself is information based, which allows us to track in uncertain adversarial environments. Our methodology is demonstrated on realistic scenes, which illustrate its robustness on challenging scenarios.

Extraction of breast lesions from ultrasound imagery: Bhattacharyya gradient flow approach

Breast cancer is one of the most commonly diagnosed neoplasms among American women and the second leading cause of death among women all over the world. In order to reduce the mortality rate and cost of treatment, early diagnosis and treatment are essential. Accurate and reliable diagnosis is required in order to ensure the most effective treatment and a second opinion is often advisable. In this paper, we address the problem of breast lesion detection from ultrasound imagery by means of active contours, whose evolution is driven by maximizing the Bhattacharyya distance1 between the probability density functions (PDFs). The proposed method was applied to ultrasound breast imagery, and the lesion boundary was obtained by maximizing the distance-based energy functional such that the maximum (optimal contour) is attained at the boundary of the potential lesion. We compared the results of the proposed method quantitatively using the Dice coefficient (similarity index)2 to well-known GrowCut segmentation method3 and demonstrated that Bhattacharyya approach outperforms GrowCut in most of the cases.

Geometry of correlation networks for studying the biology of cancer

In this note, we extend the notion of Ollivier-Ricci curvature on weighted graphs with all positive weights to the case in which the weights may also be negative. This is done by employing the Hahn-Jordan decomposition of signed measures, allowing us to extend the Earth Movers Distance to an extended class of measures. The resulting curvature will be utilized to study the robustness properties of general networks. In particular, this will be applied to certain cancer transcription networks in order to elucidate fragility (in the sense of a rate function from large deviations theory) with respect to those genes responsible for maintaining cellular homeostasis related to growth and proliferation.

Transcriptional responses to ultraviolet and ionizing radiation: An approach based on graph curvature

More than half of all cancer patients receive radiotherapy in their treatment process. However, our understanding of abnormal transcriptional responses to radiation remains poor. In this study, we employ an extended definition of Ollivier-Ricci curvature based on L1-Wasserstein distance to investigate genes and biological processes associated with ionizing radiation (IR) and ultraviolet radiation (UV) exposure using a microarray dataset. Gene expression levels were modeled on a gene interaction topology downloaded from the Human Protein Reference Database (HPRD). This was performed for IR, UV, and mock datasets, separately. The difference curvature value between IR and mock graphs (also between UV and mock) for each gene was used as a metric to estimate the extent to which the gene responds to radiation. We found that in comparison of the top 200 genes identified from IR and UV graphs, about 20∼30% genes were overlapping. Through gene ontology enrichment analysis, we found that the metabolic-related biological process was highly associated with both IR and UV radiation exposure.

Curvature Analysis of Estrogen Receptor Positive Breast Cancer Under PI3K Inhibition

In this note, we re-examine the work of Bosch et al. from a network point of view. In particular, we employ an extended defintion of Ollivier-Ricci curvature that allows us to study graphs with both positive and negative weights. This is done by utilizing a dual formulation of the Wasserstein 1-metric, allowing us to extend the Earth Mover’s Distance to signed measures. The resulting curvature may be applied study the robustness properties of general networks modelled as weighted graphs. In this note, we apply the theory to elucidate the robustness and therefore possible mechanisms of resistance of estrogen receptor positive breast cancer under PI3K inhibition.