arXiv subjects

Vartan Choulakian

Publications and source records attributed to Vartan Choulakian.

13 recordsLinked to original sources

arXiv2026

Geometric Interpretation of the Robustness of Taxicab Singular Value Decomposition of High-Dimensional Sparse Data

This paper examines the fundamental properties of projections in L2-normed (Euclidean) and L1-normed (Taxicab) spaces through two complementary frameworks : projection operators and distance minimization. The problem may be viewed as the most elementary regression problem, namely the projection of one point onto another. Its relevance stems from the fact that any SVD-like matrix decomposition can be interpreted as a pair of simultaneous regressions of the rows and columns of a data matrix within either Euclidean or Taxicab geometry. Whereas orthogonality (L2-conjugacy) is the key concept in Euclidean geometry, L1-conjugacy, characterized by the sign function, assumes an analogous role in Taxicab geometry. This contrast is further illustrated by the relationship between the classical Pythagorean theorem and its Taxicab counterpart. The primary goal of this study is to compare three decompositions : the classical singular value decomposition (SVD), the Taxicab singular value decomposition (TSVD), and the L1-min SVD. We also provide a geometric explanation for the robustness of TSVD in the analysis of extremely sparse, high-dimensional datasets by comparing the volumes of hyperspheres inscribed in hypercubes under the two geometries.

math.FA↗

arXiv2025

Scale Invariant Correspondence Analysis

Correspondence analysis is a dimension reduction method for visualization of nonnegative data sets, in particular contingency tables ; but it depends on the marginals of the data set. Two transformations of the data have been proposed to render correspondence analysis row and column scales invariant : These two kinds of transformations change the initial form of the data set into a bistochastic form. The power transorfmation applied by Greenacre (2010) has one positive parameter. While the transormation applied by Mosteller (1968) and Goodman (1996) has (I+J) positive parameters, where the raw data is row and column scaled by the Sinkhorn (RAS or ipf) algorithm to render it bistochastic. Goodman (1996) named correspondence analsis of a bistochastic matrix marginal-free correspondence analysis. We discuss these two transformations, and further generalize Mosteller-Goodman approach.

stat.ME↗

arXiv2024

Binary Trees and Taxicab Correspondence Analysis of Extremely Sparse Binary Textual Data: A Case Study

This is a case study, where Taxicab Correspondence Analysis reveals that the underlying structure of an extremely sparse binary textual data set can be represented by a binary tree, where the nodes representing clusters of words can be interpreted as topics. The textual data set represents Israel's Declaration of Independence text and 40 diverse Israeli Interviewees. The analysis provides for a compare and contrast study of textual data coming from two different sources. Furthermore, we propose an adjusted sparsity index which takes into account the size of the data table.

stat.AP↗

arXiv2024

Distortion in Correspondence Analysis and in Taxicab Correspondence Analysis: A Comparison

Distortion is a fundamental well-studied topic in dimension reduction papers, and intimately related with the underlying intrinsic dimension of a mapping of a high dimensional data set onto a lower dimension. In this paper, we study embedding distortions produced by Correspondence Analysis and its robust l1 variant Taxicab Correspondence analysis, which are visualization methods for contingency tables. For high dimensional data, distortions in Correspondence Analysis are contractions; while distortions in Taxicab Correspondence Analysis could be contractions or stretchings. This shows that Euclidean geometry is quite rigid, because of the orthogonality property; while Taxicab geometry is quite flexible, because the orthogonality property is replaced by the conjugacy property.

stat.ME↗

arXiv2023

On the choice of weights in aggregate compositional data analysis

In this paper, we distinguish between two kinds of compositional data sets: elementary and aggregate. This fact will help us to decide the choice of the weights to use in log interaction analysis of aggregate compositional vectors. We show that in the aggregate case, the underlying given data form a paired data sets composed of responses and qualitative covariates; this fact helps us to propose two approaches for analysis-visualization of data named log interaction of aggregates and aggregate of log interactions. Furthermore, we also show the first-order approximation of log interaction of a cell for different choices of the row and column weights.

stat.AP↗

arXiv2023

Notes on Correspondence Analysis of Power Transformed Data Sets

We prospect for a clear simple picture on CA of power transformed or the Box-Cox transformed data initiated since 2009 by Grenacre. We distinguish two types of data sets : strictly positive and with zeros; we concentrate on the latter.

stat.AP↗

arXiv2022

Some notes on Goodman's marginal-free correspondence analysis

In his seminal paper Goodman (1996) introduced marginal-free correspondence analysis; where his principal aim was to reconcile Pearson correlation measure with Yule's association measure in the analysis of contingency tables. We show that marginal-free correspondence analysis is a particular case of correspondence analysis with prespecified weights studied in the beginning of the 1980s by Benzécri and his students. Furthermore, we show that it is also a particular first-order approximation of logratio analysis with uniform weights. Key words: Marginal-free correspondence analysis; logratio analysis; interactions; scale invariance; taxicab singular value decomposition.

stat.AP↗

arXiv2021

Comments on "correspondence analysis makes you blind"

Collins' (2002) statement "correspondence analysis makes you blind" followed after his seriation like description of a brand attribute count data set analyzed by Whitlark and Smith (2001), who applied correspondence analysis. In this essay we comment on Collins' statement within taxicab correspondence analysis framework by simultaneously decomposing the covariance matrix and its associated density matrix, thus interpreting two interrelated maps for contingency tables : TCov map and TCA map.

stat.ME↗

arXiv2021

Quantification of intrinsic quality of a principal dimension in correspondence analysis and taxicab correspondence analysis

Collins(2002, 2011) raised a number of issues with regards to correspondence analysis (CA), such as: qualitative information in a CA map versus quantitative information in the relevant contingency table; the interpretation of a CA map is difficult and its relation with the \% of inertia (variance) explained. We tackle these issues by considering CA and taxicab CA (TCA) as a stepwise Hotelling/Tucker decomposition of the cross-covariance matrix of the row and column categories into four quadrants. The contents of this essay are: First, we review the notion of quality/quantity in multidimensional data analysis as discussed by Benzécri, who based his reflections on Aristotle. Second, we show the importance of unravelling the interrelated concepts of dependence/heterogeneity structure in a contingency table; and to picture them two maps are needed. Third, we distinguish between intrinsic and extrinsic quality of a principal dimension; the intrinsic quality is based on the signs of the residuals in the four quadrants, hence to the interpretability. Furthermore, we provide quantifications of the intrinsic quality and use them to uncover structure in particular in sparse contingency tables. Finally, we emphasize the importance of looking at the residual cross-covariance values at each iteration.

stat.ME↗

arXiv2021

Uncovering and Displaying the Coherent Groups of Rank Data by Exploratory Riffle Shuffling

Let n respondents rank order d items, and suppose that d << n. Our main task is to uncover and display the structure of the observed rank data by an exploratory riffle shuffling procedure which sequentially decomposes the n voters into a finite number of coherent groups plus a noisy group : where the noisy group represents the outlier voters and each coherent group is composed of a finite number of coherent clusters. We consider exploratory riffle shuffling of a set of items to be equivalent to optimal two blocks seriation of the items with crossing of some scores between the two blocks. A riffle shuffled coherent cluster of voters within its coherent group is essentially characterized by the following facts : a) Voters have identical first TCA factor score, where TCA designates taxicab correspondence analysis, an L1 variant of correspondence analysis ; b) Any preference is easily interpreted as riffle shuffling of its items ; c) The nature of different riffle shuffling of items can be seen in the structure of the contingency table of the first-order marginals constructed from the Borda scorings of the voters ; d) The first TCA factor scores of the items of a coherent cluster are interpreted as Borda scale of the items. We also introduce a crossing index, which measures the extent of crossing of scores of voters between the two blocks seriation of the items. The novel approach is explained on the benchmarking SUSHI data set, where we show that this data set has a very simple structure, which can also be communicated in a tabular form.

stat.ME↗

arXiv2017

Taxicab Correspondence Analysis of Sparse Contingency Tables

Visualization and interpretation of contingency tables by correspondence analysis (CA), as developed by Benzecri, has a rich structure based on Euclidean geometry. However, it is a well established fact that, often CA is very sensitive to sparse contingency tables, where we caracterize sparsity as the existence of relatively high-valued counts, rare observations discussed by Rao (1995), and zero-block structure emphasized by Novak and Bar-Hen (2005) and Greenacre (2013). In this paper, we aim to emphasize the important roles played by L1 and L2 geometries. This will be done by comparing the maps obtained by CA with the maps obtained by taxicab correspondence analysis (TCA), where TCA is a robust L1 variant of correspondence analysis. If the projections of view of both maps are quite different, we refer to this phenomenon as parallax. In astronomy, parallax means the apparent change in the position of an object as seen from two different points. In our case the two different points correspond to the two different geometries, Euclidean and Taxicab. The existence of a parallax highlights the important, but hidden, role of the underlying geometry in the interpretation of the maps obtained in multivariate data analysis. We emphasize the following fact: Only by comparing CA and TCA graphical displays, we are able to reveal the phenomenon of parallax. Examples are provided.

stat.AP↗

arXiv2015

Matrix Factorizations Based on Induced Norms

We decompose a matrix Y into a sum of bilinear terms in a stepwise manner, by considering Y as a mapping from a finite dimensional Banach space into another finite dimensional Banach space. We provide transition formulas, and represent them in a duality diagram, thus generalizing the well known duality diagram in the french school of data analysis. As an application, we introduce a family of Euclidean multidimensional scaling models.

stat.AP↗

arXiv2009

Some Numerical Results on the Rank of Generic Three-Way Arrays over R

The aim of this paper is the introduction of a new method for the numerical computation of the rank of a three-way array. We show that the rank of a three-way array over R is intimately related to the real solution set of a system of polynomial equations. Using this, we present some numerical results based on the computation of Grobner bases. Key words: Tensors; three-way arrays; Candecomp/Parafac; Indscal; generic rank; typical rank; Veronese variety; Segre variety; Grobner bases. AMS classification: Primary 15A69; Secondary 15A72, 15A18.

stat.CO↗