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**Published in:** LIPIcs, Volume 137, 34th Computational Complexity Conference (CCC 2019)

The concept of matrix rigidity was first introduced by Valiant in [Friedman, 1993]. Roughly speaking, a matrix is rigid if its rank cannot be reduced significantly by changing a small number of entries. There has been extensive interest in rigid matrices as Valiant showed in [Friedman, 1993] that rigidity can be used to prove arithmetic circuit lower bounds.
In a surprising result, Alman and Williams showed that the (real valued) Hadamard matrix, which was conjectured to be rigid, is actually not very rigid. This line of work was extended by [Dvir and Edelman, 2017] to a family of matrices related to the Hadamard matrix, but over finite fields. In our work, we take another step in this direction and show that for any abelian group G and function f:G - > {C}, the matrix given by M_{xy} = f(x - y) for x,y in G is not rigid. In particular, we get that complex valued Fourier matrices, circulant matrices, and Toeplitz matrices are all not rigid and cannot be used to carry out Valiant’s approach to proving circuit lower bounds. This complements a recent result of Goldreich and Tal [Goldreich and Tal, 2016] who showed that Toeplitz matrices are nontrivially rigid (but not enough for Valiant’s method). Our work differs from previous non-rigidity results in that those works considered matrices whose underlying group of symmetries was of the form {F}_p^n with p fixed and n tending to infinity, while in the families of matrices we study, the underlying group of symmetries can be any abelian group and, in particular, the cyclic group {Z}_N, which has very different structure. Our results also suggest natural new candidates for rigidity in the form of matrices whose symmetry groups are highly non-abelian.
Our proof has four parts. The first extends the results of [Josh Alman and Ryan Williams, 2016; Dvir and Edelman, 2017] to generalized Hadamard matrices over the complex numbers via a new proof technique. The second part handles the N x N Fourier matrix when N has a particularly nice factorization that allows us to embed smaller copies of (generalized) Hadamard matrices inside of it. The third part uses results from number theory to bootstrap the non-rigidity for these special values of N and extend to all sufficiently large N. The fourth and final part involves using the non-rigidity of the Fourier matrix to show that the group algebra matrix, given by M_{xy} = f(x - y) for x,y in G, is not rigid for any function f and abelian group G.

Zeev Dvir and Allen Liu. Fourier and Circulant Matrices Are Not Rigid. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 17:1-17:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dvir_et_al:LIPIcs.CCC.2019.17, author = {Dvir, Zeev and Liu, Allen}, title = {{Fourier and Circulant Matrices Are Not Rigid}}, booktitle = {34th Computational Complexity Conference (CCC 2019)}, pages = {17:1--17:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-116-0}, ISSN = {1868-8969}, year = {2019}, volume = {137}, editor = {Shpilka, Amir}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2019.17}, URN = {urn:nbn:de:0030-drops-108390}, doi = {10.4230/LIPIcs.CCC.2019.17}, annote = {Keywords: Rigidity, Fourier matrix, Circulant matrix} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

We introduce a simple logical inference structure we call a spanoid (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry (point-line incidences), algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation), network theory (gossip / infection processes) and coding theory. We initiate a thorough investigation of spanoids, from computational and structural viewpoints, focusing on parameters relevant to the applications areas above and, in particular, to questions regarding Locally Correctable Codes (LCCs).
One central parameter we study is the rank of a spanoid, extending the rank of a matroid and related to the dimension of codes. This leads to one main application of our work, establishing the first known barrier to improving the nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. On the one hand, we prove that the KT bound (and its more recent refinements) holds for the much more general setting of spanoid rank. On the other hand we show that there exist (random) spanoids whose rank matches these bounds. Thus, to significantly improve the known bounds one must step out of the spanoid framework.
Another parameter we explore is the functional rank of a spanoid, which captures the possibility of turning a given spanoid into an actual code. The question of the relationship between rank and functional rank is one of the main questions we raise as it may reveal new avenues for constructing new LCCs (perhaps even matching the KT bound). As a first step, we develop an entropy relaxation of functional rank to create a small constant gap and amplify it by tensoring to construct a spanoid whose functional rank is smaller than rank by a polynomial factor. This is evidence that the entropy method we develop can prove polynomially better bounds than KT-type methods on the dimension of LCCs.
To facilitate the above results we also develop some basic structural results on spanoids including an equivalent formulation of spanoids as set systems and properties of spanoid products. We feel that given these initial findings and their motivations, the abstract study of spanoids merits further investigation. We leave plenty of concrete open problems and directions.

Zeev Dvir, Sivakanth Gopi, Yuzhou Gu, and Avi Wigderson. Spanoids - An Abstraction of Spanning Structures, and a Barrier for LCCs. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dvir_et_al:LIPIcs.ITCS.2019.32, author = {Dvir, Zeev and Gopi, Sivakanth and Gu, Yuzhou and Wigderson, Avi}, title = {{Spanoids - An Abstraction of Spanning Structures, and a Barrier for LCCs}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {32:1--32:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.32}, URN = {urn:nbn:de:0030-drops-101258}, doi = {10.4230/LIPIcs.ITCS.2019.32}, annote = {Keywords: Locally correctable codes, spanoids, entropy, bootstrap percolation, gossip spreading, matroid, union-closed family} }

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**Published in:** LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L_\infty norm) with a small number of samples. We coin the term 'outlaw distributions' for such distributions since they 'defy' the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently
'smooth' functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry and from hypergraph (non)expanders.
We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.

Jop Briët, Zeev Dvir, and Sivakanth Gopi. Outlaw Distributions and Locally Decodable Codes. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{briet_et_al:LIPIcs.ITCS.2017.20, author = {Bri\"{e}t, Jop and Dvir, Zeev and Gopi, Sivakanth}, title = {{Outlaw Distributions and Locally Decodable Codes}}, booktitle = {8th Innovations in Theoretical Computer Science Conference (ITCS 2017)}, pages = {20:1--20:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-029-3}, ISSN = {1868-8969}, year = {2017}, volume = {67}, editor = {Papadimitriou, Christos H.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.20}, URN = {urn:nbn:de:0030-drops-81888}, doi = {10.4230/LIPIcs.ITCS.2017.20}, annote = {Keywords: Locally Decodable Code, VC-dimension, Incidence Geometry, Cayley Hypergraphs} }

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**Published in:** LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Kelly's theorem states that a set of n points affinely spanning C^3 must determine at least one ordinary complex line (a line passing through exactly two of the points). Our main theorem shows that such sets determine at least 3n/2 ordinary lines, unless the configuration has n-1 points in a plane and one point outside the plane (in which case there are at least n-1 ordinary lines). In addition, when at most n/2 points are contained in any plane, we prove a theorem giving stronger bounds that take advantage of the existence of lines with four and more points (in the spirit of Melchior's and Hirzebruch's inequalities). Furthermore, when the points span four or more dimensions, with at most n/2 points contained in any three dimensional affine subspace, we show that there must be a quadratic number of ordinary lines.

Abdul Basit, Zeev Dvir, Shubhangi Saraf, and Charles Wolf. On the Number of Ordinary Lines Determined by Sets in Complex Space. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{basit_et_al:LIPIcs.SoCG.2017.15, author = {Basit, Abdul and Dvir, Zeev and Saraf, Shubhangi and Wolf, Charles}, title = {{On the Number of Ordinary Lines Determined by Sets in Complex Space}}, booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-038-5}, ISSN = {1868-8969}, year = {2017}, volume = {77}, editor = {Aronov, Boris and Katz, Matthew J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.15}, URN = {urn:nbn:de:0030-drops-71883}, doi = {10.4230/LIPIcs.SoCG.2017.15}, annote = {Keywords: Incidences, Combinatorial Geometry, Designs, Polynomial Method} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

In this work we study arrangements of k-dimensional subspaces V_1,...,V_n over the complex numbers. Our main result shows that, if every pair V_a, V_b of subspaces is contained in a dependent triple (a triple V_a, V_b, V_c contained in a 2k-dimensional space), then the entire arrangement must be contained in a subspace whose dimension depends only on k (and not on n). The theorem holds under the assumption that the subspaces are pairwise non-intersecting (otherwise it is false). This generalizes the Sylvester-Gallai theorem (or Kelly's theorem for complex numbers), which proves the k=1 case. Our proof also handles arrangements in which we have many pairs (instead of all) appearing in dependent triples, generalizing the quantitative results of Barak et. al. One of the main ingredients in the proof is a strengthening of a theorem of Barthe (from the k=1 to k>1 case) proving the existence of a linear map that makes the angles between pairs of subspaces large on average. Such a mapping can be found, unless there is an obstruction in the form of a low dimensional subspace intersecting many of the spaces in the arrangement (in which case one can use a different argument to prove the main theorem).

Zeev Dvir and Guangda Hu. Sylvester-Gallai for Arrangements of Subspaces. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 29-43, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dvir_et_al:LIPIcs.SOCG.2015.29, author = {Dvir, Zeev and Hu, Guangda}, title = {{Sylvester-Gallai for Arrangements of Subspaces}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {29--43}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.29}, URN = {urn:nbn:de:0030-drops-51303}, doi = {10.4230/LIPIcs.SOCG.2015.29}, annote = {Keywords: Sylvester-Gallai, Locally Correctable Codes} }

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**Published in:** LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

We prove a new upper bound on the number of r-rich lines (lines with at least r points) in a 'truly' d-dimensional configuration of points v_1,...,v_n over the complex numbers. More formally, we show that, if the number of r-rich lines is significantly larger than n^2/r^d then there must exist a large subset of the points contained in a hyperplane. We conjecture that the factor r^d can be replaced with a tight r^{d+1}. If true, this would generalize the classic Szemeredi-Trotter theorem which gives a bound of n^2/r^3 on the number of r-rich lines in a planar configuration. This conjecture was shown to hold in R^3 in the seminal work of Guth and Katz and was also recently proved over R^4 (under some additional restrictions) by Solomon and Sharir. For the special case of arithmetic progressions (r collinear points that are evenly distanced) we give a bound that is tight up to lower order terms, showing that a d-dimensional grid achieves the largest number of r-term progressions.
The main ingredient in the proof is a new method to find a low degree polynomial that vanishes on many of the rich lines. Unlike previous applications of the polynomial method, we do not find this polynomial by interpolation. The starting observation is that the degree r-2 Veronese embedding takes r-collinear points to r linearly dependent images. Hence, each collinear r-tuple of points, gives us a dependent r-tuple of images. We then use the design-matrix method of Barak et al. to convert these 'local' linear dependencies into a global one, showing that all the images lie in a hyperplane. This then translates into a low degree polynomial vanishing on the original set.

Zeev Dvir and Sivakanth Gopi. On the Number of Rich Lines in Truly High Dimensional Sets. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 584-598, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{dvir_et_al:LIPIcs.SOCG.2015.584, author = {Dvir, Zeev and Gopi, Sivakanth}, title = {{On the Number of Rich Lines in Truly High Dimensional Sets}}, booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)}, pages = {584--598}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-83-5}, ISSN = {1868-8969}, year = {2015}, volume = {34}, editor = {Arge, Lars and Pach, J\'{a}nos}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.584}, URN = {urn:nbn:de:0030-drops-51110}, doi = {10.4230/LIPIcs.SOCG.2015.584}, annote = {Keywords: Incidences, Combinatorial Geometry, Designs, Polynomial Method, Additive Combinatorics} }