Neriman Tokcan

Neriman Tokcan
Broad Institute of MIT and Harvard · Golub Lab

Doctor of Philosophy

About

17
Publications
1,656
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105
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. I am also interested in techniques for symmetric tensor decomposition – also known as the Waring problem for forms, sum of squares optimization, Schubert polynomials, and Newton polytopes in algebraic combinatorics.
Additional affiliations
September 2019 - present
Broad Institute of MIT and Harvard
Position
  • PostDoc Position
August 2017 - September 2019
University of Michigan
Position
  • PostDoc Position
August 2017 - September 2019
University of Michigan
Position
  • PostDoc Position
Education
August 2012 - August 2017
August 2012 - May 2017
January 2011 - August 2012

Publications

Publications (17)
Article
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
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...
Preprint
Full-text available
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...
Presentation
Full-text available
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...
Article
Full-text available
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...
Article
Full-text available
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...
Preprint
Full-text available
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...
Presentation
Full-text available
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...
Poster
Full-text available
Tensor methods for the analysis of multi-dimensional, multi-modal biomedical signals.
Thesis
Full-text available
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...
Conference Paper
Full-text available
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...
Presentation
Full-text available
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...

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