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J. M. Landsberg

Publications and source records attributed to J. M. Landsberg.

At least 37 records · Page 2Linked to original sources
arXiv2016

The method of shifted partial derivatives cannot separate the permanent from the determinant

The method of shifted partial derivatives was used to prove a super-polynomial lower bound on the size of depth four circuits needed to compute the permanent. We show that this method alone cannot prove that the padded permanent $\ell^{n-m} perm_m$ cannot be realized inside the $GL_{n^2}$-orbit closure of the determinant $ det_n$ when $n>2m^2+2m$. Our proof relies on several simple degenerations of the determinant polynomial, Macaulay's theorem that gives a lower bound on the growth of an ideal, and a lower bound estimate from Gupta et. al. regarding the shifted partial derivatives of the determinant.

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arXiv2016

Abelian Tensors

We analyze tensors in the tensor product of three m-dimensional vector spaces satisfying Strassen's equations for border rank m. Results include: two purely geometric characterizations of the Coppersmith-Winograd tensor, a reduction to the study of symmetric tensors under a mild genericity hypothesis, and numerous additional equations and examples. This study is closely connected to the study of the variety of m-dimensional abelian subspaces of the space of endomorphisms of an m-dimensional vector space, and the subvariety consisting of the Zariski closure of the variety of maximal tori, called the variety of reductions.

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arXiv2016

An explicit description of the irreducible components of the set of matrix pencils with bounded normal rank

The set of mxn singular matrix pencils with normal rank at most r is an algebraic set with r+1 irreducible components. These components are the closure of the orbits (under strict equivalence) of r+1 matrix pencils which are in Kronecker canonical form. In this paper, we provide a new explicit description of each of these irreducible components which is a parametrization of each component. Therefore one can explicitly construct any pencil in each of these components. The new description of each of these irreducible components consists of the sum of r rank-1 matrix pencils, namely, a column polynomial vector of degree at most 1 times a row polynomial vector of degree at most 1, where we impose one of these two vectors to have degree zero. The number of row vectors with zero degree determines each irreducible component.

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arXiv2016

On the geometry of border rank algorithms for matrix multiplication and other tensors with symmetry

We establish basic information about border rank algorithms for the matrix multiplication tensor and other tensors with symmetry. We prove that border rank algorithms for tensors with symmetry (such as matrix multiplication and the determinant polynomial) come in families that include representatives with normal forms. These normal forms will be useful both to develop new efficient algorithms and to prove lower complexity bounds. We derive a border rank version of the substitution method used in proving lower bounds for tensor rank. We use this border-substitution method and a normal form to improve the lower bound on the border rank of matrix multiplication by one, to 2n^2- n+1. We also point out difficulties that will be formidable obstacles to future progress on lower complexity bounds for tensors because of the "wild" structure of the Hilbert scheme of points.

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arXiv2015

An introduction to geometric complexity theory

I survey methods from differential geometry, algebraic geometry and representation theory relevant for the permanent v. determinant problem from computer science, an algebraic analog of the P v. NP problem.

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arXiv2013

Geometric Complexity Theory: an introduction for geometers

This article is a survey of recent developments in, and a tutorial on, the approach to P v. NP and related questions called Geometric Complexity Theory (GCT). It is written to be accessible to graduate students. Numerous open questions in algebraic geometry and representation theory relevant for GCT are presented.

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arXiv2013

New lower bounds for the rank of matrix multiplication

The rank of the matrix multiplication operator for nxn matrices is one of the most studied quantities in algebraic complexity theory. I prove that the rank is at least n^2-o(n^2). More precisely, for any integer p\leq n -1, the rank is at least (3- 1/(p+1))n^2-(1+2p\binom{2p}{p-1})n. The previous lower bound, due to Blaser, was 5n^2/2-3n (the case p=1). The new bounds improve Blaser's bound for all n>84. I also prove lower bounds for rectangular matrices significantly better than the the previous bound.

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arXiv2013

Explicit tensors of border rank at least 2n-1

For odd n, I write down tensors in C^n\otimes C^n\otimes C^n of border rank 2n-1, showing the non-triviality of the Young-flattening equations of Landsberg-Ottaviani. I also study the border rank of the tensors of Alexeev et. al., showing the tensors their tensors T_{2^k}, despite having rank equal to 2^{k+1}-1, have border rank equal to 2^k, the minimum of any concise tensor. I also study the equations of Griesser.

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arXiv2013

Equations for lower bounds on border rank

We present new methods for determining polynomials in the ideal of the variety of bilinear maps of border rank at most r. We apply these methods to several cases including the case r = 6 in the space of bilinear maps C^4 x C^4 -> C^4. This space of bilinear maps includes the matrix multiplication operator M_2 for two by two matrices. We show these newly obtained polynomials do not vanish on the matrix multiplication operator M_2, which gives a new proof that the border rank of the multiplication of 2 x 2 matrices is seven. Other examples are considered along with an explanation of how to implement the methods.

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arXiv2013

New lower bounds for the border rank of matrix multiplication

The border rank of the matrix multiplication operator for n by n matrices is a standard measure of its complexity. Using techniques from algebraic geometry and representation theory, we show the border rank is at least 2n^2-n. Our bounds are better than the previous lower bound (due to Lickteig in 1985) of 3/2 n^2+ n/2 -1 for all n>2. The bounds are obtained by finding new equations that bilinear maps of small border rank must satisfy, i.e., new equations for secant varieties of triple Segre products, that matrix multiplication fails to satisfy.

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arXiv2012

On the third secant variety

We determine normal forms and ranks of tensors of border rank at most three. We present a differential-geometric analysis of limits of secant planes in a more general context. In particular there are at most four types of points on limiting trisecant planes for cominuscule varieties such as Grassmannians. We also show the singular locus of the first two secant varieties of all triple Segre products has codimension at least two.

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arXiv2012

Ranks of tensors and a generalization of secant varieties

We introduce subspace rank as a tool for studying ranks of tensors and X-rank more generally. We derive a new upper bound for the rank of a tensor and determine the ranks of partially symmetric tensors in C^2 \otimes C^b \otimes C^b. We review the literature from a geometric perspective.

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arXiv2012

Determinantal equations for secant varieties and the Eisenbud-Koh-Stillman conjecture

We address special cases of a question of Eisenbud on the ideals of secant varieties of Veronese re-embeddings of arbitrary varieties. Eisenbud's question generalizes a conjecture of Eisenbud, Koh and Stillman (EKS) for curves. We prove that set-theoretic equations of small secant varieties to a high degree Veronese re-embedding of a smooth variety are determined by equations of the ambient Veronese variety and linear equations. However this is false for singular varieties, and we give explicit counter-examples to the EKS conjecture for singular curves. The techniques we use also allow us to prove a gap and uniqueness theorem for symmetric tensor rank. We put Eisenbud's question in a more general context about the behaviour of border rank under specialisation to a linear subspace, and provide an overview of conjectures coming from signal processing and complexity theory in this context.

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arXiv2012

Padded polynomials, their cousins, and geometric complexity theory

We establish basic facts about the varieties of homogeneous polynomials divisible by powers of linear forms, and explain consequences for geometric complexity theory. This includes quadratic set-theoretic equations, a description of the ideal in terms of the kernel of a linear map that generalizes the Foulkes-Howe map, and an explicit description of the coordinate ring of the normalization. We also prove asymptotic injectivity of the Foulkes-Howe map.

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arXiv2012

On the geometry of tensor network states

We answer a question of L. Grasedyck that arose in quantum information theory, showing that the limit of tensors in a space of tensor network states need not be a tensor network state. We also give geometric descriptions of spaces of tensor networks states corresponding to trees and loops. Grasedyck's question has a surprising connection to the area of Geometric Complexity Theory, in that the result is equivalent to the statement that the boundary of the Mulmuley-Sohoni type variety associated to matrix multiplication is strictly larger than the projections and re-labelings of matrix multiplication. Tensor Network States are also related to graphical models in algebraic statistics.

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