Tensors: Geometry and Applications

By J. M. Landsberg. 2011

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In Computational Mathematics

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A literature survey of low‐rank tensor approximation techniques (2013, neutral): “In general, this statement does not hold for tensor networks containing cycles [177, 178].”
Tensor product methods and entanglement optimization for ab initio quantum chemistry (2014, background): “Thus (116) can be solved approximately by projecting f(U) on the tangent space TUMr, F (U) := PTUf(U) (119)”
A comparison of different notions of ranks of symmetric tensors (2012, background): “uce all the diﬀerent notions of rank of a homogeneous polynomial f ∈ Sd, that we will use all along the paper. 2.1. Rank and border rank. The following deﬁnition is nowadays a classical one, see e.g. [27] and references therein. Deﬁnition 2.1 (Rank). Let σ0,d r ⊂ P(Sd) be the set of projective class”
On the geometry of tensor network states (2011, background): “This type of bundle construction is standard, see [10, 16].”
Most Tensor Problems Are NP-Hard (2009, background): “It is still not known whether a symmetric tensor’s symmetric rank is always its rank (this is the Comon Conjecture [Landsberg 2012]; see also [Reznick 2013]), although the best symmetric rank-1 approximation and the best rank-1 approximation coincide (see Section 10).”
Learning algebraic varieties from samples (2018, neutral): “200 m 110 m 101 m 300 m 210 m 201 m 120 m 111 m 102 m 010 m 110 m 020 m 011 m 210 m 120 m 111 m 030 m 021 m 012 m 001 m 101 m 011 m 002 m 201 m 111 m 102 m 021 m 012 m 003 3 7 5: See Landsberg’s book [37] for an introduction to the geometry of tensors and their rank. Example 2.8. In distance geometr”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “Other presentations of this topic from different points of view can be found in [24, 23, 25, 26].”
Identifiability of homogeneous polynomials and Cremona transformations (2016, background): “sition yields a canonical form of a general polynomial. In modern terminology this is called identiﬁability. Applications of identiﬁability range from BSS to Phylogenetic and Algebraic statistic, see [Lan] for an account of applications, and it is studied for both polynomials and tensors, see for in”
How hard is the tensor rank (2016, background): “We note that the rank decompositions have proved to be useful for studies in statistics [34], signal processing [26], complexity of computation [39], psychometrics [8], linguistics [38], chemometrics [10], and we refer the reader to the monograph [24] for a more complete survey of applications.”
Model complexity of deep learning: a survey (2021, methodology): “Specifically, polynomial networks with h = 2 are connected to CP tensor decompositions [56], and deep polynomial networks are connected to an iterated tensor decomposition [65].”
Stratification of the fourth secant variety of Veronese varieties via the symmetric rank (2010, background): “imes bX m,d (P) is called either “the secant rank of P” or the “symmetric border rank of P”. A natural question arising from the applications (see for example [1], [16], [11], [15], [26], [13], [20], [23]) is: Question 1. GivenasymmetrictensorT∈ SdV(orahomogeneouspolynomialF∈ K[x0,...,xm]d), which i”
Homotopy techniques for tensor decomposition and perfect identifiability (2014, neutral): “ This variety is identiﬁed with the cone over the rth secant variety to the Segre 4 J.D.HAUENSTEIN,L.OEDING,G.OTTAVIANI,ANDA.J.SOMMESE variety P(Cn1) × ···× P(Cn d) of decomposable tensors, e.g., see [16,49]. In particular, it is meaningful to speak about a general tensor of rank r. Throughout this ”
Tensor Decomposition for Signal Processing and Machine Learning (2016, background): “It has been shown that this symmetric rank equals d ( I+N−1 N ) /Ie almost surely except in certain very special cases, see [23].”
Pathologies on the Hilbert scheme of points (2018, background): “Hilbert schemes of points also appear prominently in the study of secant and cactus varieties and in algebraic complexity [Lan12].”
Tensor Decompositions for Signal Processing Applications: From two-way to multiway component analysis (2014, background): “An important difference between matrices and tensors is that the existence of a best rank-R approximation of a tensor of rank greater than R is not guaranteed [20,62] since the set of tensors whose rank is at most R is not closed.”
On generic identifiability of symmetric tensors of subgeneric rank (2015, background): “As general sources on tensor decomposition, we refer to [13, 16, 23, 25, 34].”