# Neriman TokcanUniversity of Massachusetts Boston | UMB · Mathematics

Neriman Tokcan

Doctor of Philosophy

## About

18

Publications

2,438

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330

Citations

Introduction

I work on the generalization of matrix-based compression, noise elimination, and dimension reduction methods to higher-dimensions for the analysis of multi-modal, multi-dimensional data. I explore applications in bioinformatics and genomics.
Currently, I work on the formulation of the novel, mathematically sound tensor-based frameworks, and the development of computational tools to model tumor microenvironments.

Additional affiliations

September 2019 - present

August 2017 - September 2019

**University of Michigan**

Position

- PostDoc Position

August 2017 - September 2019

Education

August 2012 - August 2017

August 2012 - May 2017

January 2011 - August 2012

## Publications

Publications (18)

Suppose $f(x,y)$ is a binary form of degree $d$ with coefficients in a field $K \subseteq \mathbb C$. The $K$-rank of $f$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We prove that for $d \ge 5$, there always exists a form of degree $d$ with at least three different ranks over various fi...

The $K$-rank of a binary form $f$ in $K[x,y],~K\subseteq \mathbb{C},$ is the smallest number of $d$-th powers of linear forms over $K$ of which $f$ is a $K$-linear combination. We provide lower bounds for the $\mathbb{C}$-rank (Waring rank) and for the $\mathbb{R}$-rank (real Waring rank) of binary forms depending on their factorization. We complet...

A polynomial has saturated Newton polytope (SNP) if every lattice point of the convex hull of its exponent vectors corresponds to a monomial. We compile instances of SNP in algebraic combinatorics (some with proofs, others conjecturally): skew Schur polynomials; symmetric polynomials associated to reduced words, Redfield--Polya theory, Witt vectors...

We develop algebraic methods for computations with tensor data. We give 3 applications: extracting features that are invariant under the orthogonal symmetries in each of the modes, approximation of the tensor spectral norm, and amplification of low rank tensor structure. We introduce colored Brauer diagrams, which are used for algebraic computation...

Abstract: With the recent advances in information technology, high-dimensional
datasets are commonplace. When working in higher-dimensional spaces a set of new
challenges arise, known as the "curse of dimensionality". The curse, not only dictates
that computational complexity explodes in memory and time, but also that the problem
of noise and missi...

Tensor factorizations (TF) are powerful tools for the efficient representation and analysis of multidimensional data. However, classic TF methods based on maximum likelihood estimation underperform when applied to zero-inflated count data, such as single-cell RNA sequencing (scRNA-seq) data. Additionally, the stochasticity inherent in TFs results i...

Charting an organs’ biological atlas requires us to spatially resolve the entire single-cell transcriptome, and to relate such cellular features to the anatomical scale. Single-cell and single-nucleus RNA-seq (sc/snRNA-seq) can profile cells comprehensively, but lose spatial information. Spatial transcriptomics allows for spatial measurements, but...

Patients recovering from cardiovascular surgeries may develop life-threatening complications such as hemodynamic decompensation, making the monitoring of patients for such complications an essential component of postoperative care. However, this need has given rise to an inexorable increase in the number and modalities of data points collected, mak...

Charting a biological atlas of an organ, such as the brain, requires us to spatially-resolve whole transcriptomes of single cells, and to relate such cellular features to the histological and anatomical scales. Single-cell and single-nucleus RNA-Seq (sc/snRNA-seq) can map cells comprehensively, but relating those to their histological and anatomica...

A tensor is a multi-way extension of a matrix. Tensor decomposition methods can be considered as generalization of matrix methods (such as PCA, Non-Negative Matrix Factorization) to higher dimensions. In this talk, I give a brief introduction to tensors and generalization of matrix methods. I provide applications of tensor decomposition in differen...

Tensor methods for the analysis of multi-dimensional, multi-modal biomedical signals.

Suppose f (x, y) is a binary form of degree d with coefficients in a field K ⊆ C. The K-rank of f is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d ≥ 5, there always exists a form of degree d with at least three different ranks over various fields. We also study the relation betwe...

In the last decade there has been a steady uptrend in the popularity of embedded multi-core platforms. This represents a turning point in the theory and implementation of real-time systems. From a real-time standpoint, however, the extensive sharing of hardware resources (e.g. caches, DRAM subsystem, I/O channels) represents a major source of unpre...

Suppose f(x, y) is a binary form of degree d with coefficients in a field K ⊆ C. The K-rank of f, LK(f), is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d ≥ 5, there always exists a form of degree d with at least three different ranks over various fields. We also find lower bounds...