Publications
16 results found
Cao Y, Monod A, Vlontzos A, et al., 2023, Topological information retrieval with dilation-invariant bottleneck comparative measures, Information and Inference: a Journal of the IMA, Vol: 12, Pages: 1964-1996, ISSN: 2049-8772
Appropriately representing elements in a database so that queries may be accurately matched is a central task in information retrieval; recently, this has been achieved by embedding the graphical structure of the database into a manifold in a hierarchy-preserving manner using a variety of metrics. Persistent homology is a tool commonly used in topological data analysis that is able to rigorously characterize a database in terms of both its hierarchy and connectivity structure. Computing persistent homology on a variety of embedded datasets reveals that some commonly used embeddings fail to preserve the connectivity. We show that those embeddings which successfully retain the database topology coincide in persistent homology by introducing two dilation-invariant comparative measures to capture this effect: in particular, they address the issue of metric distortion on manifolds. We provide an algorithm for their computation that exhibits greatly reduced time complexity over existing methods. We use these measures to perform the first instance of topology-based information retrieval and demonstrate its increased performance over the standard bottleneck distance for persistent homology. We showcase our approach on databases of different data varieties including text, videos and medical images.
Améndola C, Monod A, 2023, An invitation to tropical Alexandrov curvature, Algebraic Statistics, ISSN: 2693-2997
We study Alexandrov curvature in the tropical projective torus with respectto the tropical metric, which has been useful in various statistical analyses,particularly in phylogenomics. Alexandrov curvature is a generalization ofclassical Riemannian sectional curvature to more general metric spaces; it isdetermined by a comparison of triangles in an arbitrary metric space tocorresponding triangles in Euclidean space. In the polyhedral setting oftropical geometry, triangles are a combinatorial object, which adds acombinatorial dimension to our analysis. We study the effect that the triangletypes have on curvature, and what can be revealed about these types from thecurvature. We find that positive, negative, zero, and undefined Alexandrovcurvature can exist concurrently in tropical settings and that there is a tightconnection between triangle combinatorial type and curvature. Our results areestablished both by proof and computational experiments, and shed light on theintricate geometry of the tropical projective torus. In this context, wediscuss implications for statistical methodologies which admit inherentgeometric interpretations. This paper is dedicated to Bernd Sturmfels on the occasion of his 60thbirthday.
Lin B, Monod A, Yoshida R, 2022, Tropical Geometric Variation of Tree Shapes, DISCRETE & COMPUTATIONAL GEOMETRY, Vol: 68, Pages: 817-849, ISSN: 0179-5376
- Author Web Link
- Cite
- Citations: 2
Tramontano D, Monod A, Drton M, 2022, Learning Linear Non-Gaussian Polytree Models, 38th Conference on Uncertainty in Artificial Intelligence, Publisher: PMLR, Pages: 1-10
In the context of graphical causal discovery, we adapt the versatileframework of linear non-Gaussian acyclic models (LiNGAMs) to propose newalgorithms to efficiently learn graphs that are polytrees. Our approachcombines the Chow--Liu algorithm, which first learns the undirected treestructure, with novel schemes to orient the edges. The orientation schemesassess algebraic relations among moments of the data-generating distributionand are computationally inexpensive. We establish high-dimensional consistencyresults for our approach and compare different algorithmic versions innumerical experiments.
Lin B, Monod A, Yoshida R, 2022, Tropical Geometric Variation of Phylogenetic Tree Shapes, Discrete and Computational Geometry: an international journal of mathematics and computer science, ISSN: 0179-5376
We study the behavior of phylogenetic tree shapes in the tropical geometricinterpretation of tree space. Tree shapes are formally referred to as treetopologies; a tree topology can also be thought of as a tree combinatorialtype, which is given by the tree's branching configuration and leaf labeling.We use the tropical line segment as a framework to define notions of varianceas well as invariance of tree topologies: we provide a combinatorial searchtheorem that describes all tree topologies occurring along a tropical linesegment, as well as a setting under which tree topologies do not change along atropical line segment. Our study is motivated by comparison to the moduli spaceendowed with a geodesic metric proposed by Billera, Holmes, and Vogtmann(referred to as BHV space); we consider the tropical geometric setting as analternative framework to BHV space for sets of phylogenetic trees. We give analgorithm to compute tropical line segments which is lower in computationalcomplexity than the fastest method currently available for BHV geodesics andshow that its trajectory behaves more subtly: while the BHV geodesic traversesthe origin for vastly different tree topologies, the tropical line segmentbypasses it.
Monod A, Sigbeku J, Saucan E, 2022, Curved Markov Chain Monte Carlo for network learning, 0th International Conference on Complex Networks and their Applications, Publisher: Springer Verlag, Pages: 461-473, ISSN: 1860-949X
We present a geometrically enhanced Markov chain Monte Carlo sampler for networks based on a discretecurvature measure defined on graphs. Specifically, we incorporate the concept of graph Forman curvatureinto sampling procedures on both the nodes and edges of a network explicitly, via the transition probabilityof the Markov chain, as well as implicitly, via the target stationary distribution, which gives a novel, curvedMarkov chain Monte Carlo approach to learning networks. We show that integrating curvature into thesampler results in faster convergence to a wide range of network statistics demonstrated on deterministicnetworks drawn from real-world data.
Lee W, Li W, Lin B, et al., 2021, Tropical Optimal Transport and Wasserstein Distances, Information Geometry, ISSN: 2511-2481
We study the problem of optimal transport in tropical geometry and define the Wasserstein-$p$ distances in the continuous metric measure space setting of the tropical projective torus. We specify the tropical metric -- a combinatorial metric that has been used to study of the tropical geometric space of phylogenetic trees -- as the ground metric and study the cases of $p=1,2$ in detail. The case of $p=1$ gives an efficient computation of the infinitely-many geodesics on the tropical projective torus, while the case of $p=2$ gives a form for Fr\'{e}chet means and a general inner product structure. Our results also provide theoretical foundations for geometric insight a statistical framework in a tropical geometric setting. We construct explicit algorithms for the computation of the tropical Wasserstein-1 and 2 distances and prove their convergence. Our results provide the first study of the Wasserstein distances and optimal transport in tropical geometry. Several numerical examples are provided.
Gartrell-Corrado RD, Chen AX, Rizk EM, et al., 2020, Linking transcriptomic and imaging data defines features of a favorable tumor immune microenvironment and identifies a combination biomarker for primary melanoma, Cancer Research, Vol: 80, Pages: 1078-1087, ISSN: 0008-5472
Patients with resected stage II-III melanoma have approximately a 35% chance of death from their disease. A deeper understanding of the tumor immune microenvironment (TIME) is required to stratify patients and identify factors leading to therapy resistance. We previously identified that the melanoma immune profile (MIP), an IFN-based gene signature, and the ratio of CD8+ cytotoxic T lymphocytes (CTL) to CD68+ macrophages both predict disease-specific survival (DSS). Here, we compared primary with metastatic tumors and found that the nuclei of tumor cells were significantly larger in metastases. The CTL/macrophage ratio was significantly different between primary tumors without distant metastatic recurrence (DMR) and metastases. Patients without DMR had higher degrees of clustering between tumor cells and CTLs, and between tumor cells and HLA-DR+ macrophages, but not HLA-DR− macrophages. The HLA-DR− subset coexpressed CD163+CSF1R+ at higher levels than CD68+HLA-DR+ macrophages, consistent with an M2 phenotype. Finally, combined transcriptomic and multiplex data revealed that densities of CD8 and M1 macrophages correlated with their respective cell phenotype signatures. Combination of the MIP signature with the CTL/macrophage ratio stratified patients into three risk groups that were predictive of DSS, highlighting the potential use of combination biomarkers for adjuvant therapy.
Crawford L, Monod A, Chen AX, et al., 2019, Predicting clinical outcomes in Glioblastoma: an application of topological and functional data analysis, Journal of the American Statistical Association, Vol: 115, Pages: 1139-1150, ISSN: 0162-1459
Glioblastoma multiforme (GBM) is an aggressive form of human brain cancer that is under active study in the field of cancer biology. Its rapid progression and the relative time cost of obtaining molecular data make other readily available forms of data, such as images, an important resource for actionable measures in patients. Our goal is to use information given by medical images taken from GBM patients in statistical settings. To do this, we design a novel statistic—the smooth Euler characteristic transform (SECT)—that quantifies magnetic resonance images of tumors. Due to its well-defined inner product structure, the SECT can be used in a wider range of functional and nonparametric modeling approaches than other previously proposed topological summary statistics. When applied to a cohort of GBM patients, we find that the SECT is a better predictor of clinical outcomes than both existing tumor shape quantifications and common molecular assays. Specifically, we demonstrate that SECT features alone explain more of the variance in GBM patient survival than gene expression, volumetric features, and morphometric features. The main takeaways from our findings are thus 2-fold. First, they suggest that images contain valuable information that can play an important role in clinical prognosis and other medical decisions. Second, they show that the SECT is a viable tool for the broader study of medical imaging informatics. Supplementary materials for this article, including a standardized description of the materials available for reproducing the work, are available as an online supplement.
Gartrell R, Chen A, Rizk E, et al., 2019, Combining transcriptomic and tissue-based immune biomarkers to improve recurrence prediction in stage II-III melanoma, Journal for ImmunoTherapy of Cancer, ISSN: 2051-1426
Monod A, Kališnik S, Patino-Galindo JÁ, et al., 2019, Tropical sufficient statistics for persistent homology, SIAM Journal on Applied Algebra and Geometry, Vol: 3, Pages: 337-371, ISSN: 2470-6566
We show that an embedding in Euclidean space based on tropical geometry generates stable sufficient statistics for barcodes. In topological data analysis, barcodes are multiscale summaries of algebraic topological characteristics that capture the “shape” of data; however, in practice, they have complex structures that make them difficult to use in statistical settings. The sufficiency result presented in this work allows for classical probability distributions to be assumed on the tropical geometric representation of barcodes. This makes a variety of parametric statistical inference methods accessible to barcodes, all while maintaining their initial interpretations. More specifically, we show that exponential family distributions may be assumed and that likelihood functions for persistent homology may be constructed. We conceptually demonstrate sufficiency and illustrate its utility in persistent homology dimensions 0 and 1 with concrete parametric applications to human immunodeficiency virus and avian influenza data.
Chen AX, Gartrell R, Marks DK, et al., 2019, Abstract B005: Linking transcriptomic and imaging features of the melanoma tumor microenvironment, Publisher: American Association for Cancer Research (AACR), Pages: B005-B005, ISSN: 2326-6066
<jats:title>Abstract</jats:title> <jats:p>Background: While immunotherapy has demonstrated success in melanoma, a deeper understanding of the heterogeneous tumor microenvironment is needed for stratifying patients for treatment. Technologies such as quantitative multiplex immunofluorescence (qmIF) imaging and transcriptomic profiling both have the potential to provide such insights. Using these tools, we had previously shown that immune cellular compositions and bulk gene expression in tumors are each related to patient survival. However, the connection between these modalities and their impact on prognosis is not well-understood. Here, we investigate the link between the microenvironmental composition of immune cells, such as cytotoxic T lymphocytes (CTLs) and macrophages, with observed transcriptomic signatures. Furthermore, we uncover spatial correlations in cellular positioning, which supports a mechanistic basis underlying these relationships. Methods: QmIF imaging was obtained from 104 stage II-III primary melanomas from Columbia University Irving Medical Center (CUIMC), which included staining for CTLs (CD3 and CD8), macrophages (CD68), and tumor cells (SOX10). HLA-DR and Ki67 were also stained in order to assess activation and proliferation of immune and tumor cells. Visualization, cell segmentation, and phenotyping were performed using inForm software within Mantra workstation. Spatial relationships between cells were quantified via the inhomogeneous pair correlation function (PCF), which compares the observed probability of two cell types being separated by a given distance to that expected by chance. Disease-specific survival status was known for 64 patients, while mRNA expression for 63 immune-related genes was obtained via NanoString for 53 patients. We assessed the similarity of each sample’s gene expression profile to the cellular subtype signatures from LM22, the reference standard for CIBERSORT. Results: The pro
Gartrell RD, Chen A, Marks DK, et al., 2018, Abstract 2091: Quantitative compositional and spatial analysis of tumor microenvironment (TME) in primary melanoma, Publisher: American Association for Cancer Research (AACR), Pages: 2091-2091, ISSN: 0008-5472
<jats:title>Abstract</jats:title> <jats:p>Background: Stratifying melanoma patients for adjuvant immunotherapy trials has quickly become an urgent need. Tumor-infiltrating lymphocyte (TIL) analysis is predictive but insufficiently precise for clinical application. A novel pathology method, quantitative multiplex immunofluorescence (qmIF), allows for complex analysis of the tumor microenvironment (TME) for development of new biomarkers. Given that CD3+CD8+ cytotoxic lymphocytes (CTLs) promote antitumor immunity while CD68+ macrophages (Mϕ) impair immunity, we hypothesized that precise quantification and spatial analysis of Mϕ and CTLs would correlate with survival in melanoma.</jats:p> <jats:p>Methods: We applied qmIF to 104 stage II/III primary melanomas from Columbia University Medical Center (CUMC); known cause of death is available for 64 patients. 5-µm slides were stained using qmIF for DAPI, CD3, CD8, CD68, SOX10, HLA-DR and Ki67. Tumor areas were preselected by a dermatopathologist and visualized using Mantra. Tissue and cell segmentation, multiparameter immune phenotyping, and quantitative spatial analysis (qSA) were performed using inForm. Local spatial analysis was implemented by partitioning images into 100-µm square windows. The interquartile range of cellular ratios among windows was used as the measure of dispersion.</jats:p> <jats:p>Results: We find high CTL and low Mϕ density in stroma (p=0.0038, p=0.0006) correlate with disease-specific survival (DSS). Correlation was strongest for stromal HLA-DR+ CTLs (p=0.0005). CTL distance to HLA-DR- Mϕ associates with poor DSS (p=0.0016), while distance to Ki67- tumor cells associates inversely with DSS (p=0.0006). Low CTL/ Mϕ ratio in stroma confers a hazard ratio (HR) of 3.719 for death from melanoma and correlates with shortened overall survival (OS) in the complete 104-patient cohort by Cox analysis (p=0.009). HLA
Gartrell RD, Marks DK, Hart TD, et al., 2018, Quantitative analysis of immune infiltrates in primary melanoma, Cancer Immunology Research, Vol: 6, Pages: 481-493, ISSN: 2326-6066
Novel methods to analyze the tumor microenvironment (TME) are urgently needed to stratify melanoma patients for adjuvant immunotherapy. Tumor-infiltrating lymphocyte (TIL) analysis, by conventional pathologic methods, is predictive but is insufficiently precise for clinical application. Quantitative multiplex immunofluorescence (qmIF) allows for evaluation of the TME using multiparameter phenotyping, tissue segmentation, and quantitative spatial analysis (qSA). Given that CD3+CD8+ cytotoxic lymphocytes (CTLs) promote antitumor immunity, whereas CD68+ macrophages impair immunity, we hypothesized that quantification and spatial analysis of macrophages and CTLs would correlate with clinical outcome. We applied qmIF to 104 primary stage II to III melanoma tumors and found that CTLs were closer in proximity to activated (CD68+HLA-DR+) macrophages than nonactivated (CD68+HLA-DR−) macrophages (P < 0.0001). CTLs were further in proximity from proliferating SOX10+ melanoma cells than nonproliferating ones (P < 0.0001). In 64 patients with known cause of death, we found that high CTL and low macrophage density in the stroma (P = 0.0038 and P = 0.0006, respectively) correlated with disease-specific survival (DSS), but the correlation was less significant for CTL and macrophage density in the tumor (P = 0.0147 and P = 0.0426, respectively). DSS correlation was strongest for stromal HLA-DR+ CTLs (P = 0.0005). CTL distance to HLA-DR− macrophages associated with poor DSS (P = 0.0016), whereas distance to Ki67− tumor cells associated inversely with DSS (P = 0.0006). A low CTL/macrophage ratio in the stroma conferred a hazard ratio (HR) of 3.719 for death from melanoma and correlated with shortened overall survival (OS) in the complete 104 patient cohort by Cox analysis (P = 0.009) and merits further development as a biomarker for clinical application.
Monod A, 2014, Random Effects Modeling and the Zero-Inflated Poisson Distribution, Communications in Statistics - Theory and Methods, Vol: 43, Pages: 664-680, ISSN: 0361-0926
Monod A, 2011, Generalized Estimating Equations for Zero-Inflated Spatial Count Data, Procedia Environmental Sciences, Vol: 7, Pages: 281-286, ISSN: 1878-0296
This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.