We resolve the ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo, and Willsky up to a vanishing factor. Concretely, for m independent Gaussian points in dimension d, we show that with high probability, for m ≤ (1-o_d(1)) · d^2/4, there exists a centered ellipsoid passing through all m points; for m≥ (1+o_d(1) )· d^2/4, no such ellipsoid exists. This confirms that the ellipsoid fitting problem has a sharp phase transition at d^2/4.

For an arbitrary family of predicates ℱ⊆{0,1}^[q]^k and any ε> 0, we prove a single-pass, linear-space streaming lower bound against the gap promise problem of distinguishing instances of Max-CSP(ℱ) with at most β+ε fraction of satisfiable constraints from instances of with at least γ-ε fraction of satisfiable constraints, whenever Max-CSP(ℱ) admits a (γ,β)-integrality gap instance for the basic LP. This subsumes the linear-space lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy (STOC 2022), which applies only to a special subclass of CSPs with linear-algebraic structure. (Their result itself generalizes work of Kapralov and Krachun (STOC 2019) for Max-CUT.) Our approach identifies the right “analytic” analogues of previously-used linear-algebraic conditions; this yields substantial simplifications while capturing a much larger class of problems. Our lower bound is essentially optimal for single-pass streaming, since: (1) All CSPs admit (1-ε)-approximations in quasilinear space, and (2) sublinear-space streaming algorithms can simulate the LP (on bounded-degree instances), giving approximation algorithms when integrality gap instances do not exist. The starting point for our lower bound is a reduction from a "distributional implicit hidden partition” problem defined by Fei, Minzer, and Wang (STOC 2026) in the context of multi-pass streaming. Our result is an analogue of theirs in the single-pass setting, where we obtain a much stronger (and tight) space lower bound.

We give a new framework based on graph regularity lemmas, for list decoding and list recovery of codes based on spectral expanders. Using existing algorithms for computing regularity decompositions of sparse graphs in (randomized) near-linear time, and appropriate choices for the constant-sized inner/base codes, we prove the following: - Expander-based codes constructed using the distance amplification technique of Alon, Edmonds and Luby [FOCS 1995] can be list decoded to capacity in near-linear time. By known results, the output list is optimal up to constant factors. - The same codes of Alon, Edmonds and Luby, can also be list recovered to capacity in near-linear time, with constant-sized output lists. - The Tanner code construction of Sipser and Spielman [IEEE Trans. Inf. Theory 1996] can be list decoded to its distance in near-linear time, with constant-sized output lists. Our results imply novel combinatorial as well as algorithmic bounds for each of the above explicit constructions. All of these bounds are obtained via combinatorial rigidity phenomena, proved using (weak) graph regularity. The regularity framework allows us to lift the list decoding and list recovery properties for the local base codes, to the global codes obtained via the above constructions.

We present a new method for obtaining norm bounds for random matrices, where each entry is a low-degree polynomial in an underlying set of independent real-valued random variables. Such matrices arise in a variety of settings in the analysis of spectral and optimization algorithms, which require understanding the spectrum of a random matrix depending on data obtained as independent samples. Using ideas of decoupling and linearization from analysis, we show a simple way of expressing norm bounds for such matrices, in terms of matrices of lower-degree polynomials corresponding to derivatives. Iterating this method gives a simple bound with an elementary proof, which can recover many bounds previously required more involved techniques.

We identify a connection between the approximability of CSPs in two models: (i) sublinear space streaming algorithms, and (ii) the basic LP relaxation. We show that whenever the basic LP admits an integrality gap, there is an Ω(√(n))-space sketching lower bound. We also show that all existing linear space streaming lower bounds for Max-CSPs can be lifted to integrality gap instances for basic LPs. For bounded-degree graphs, by combining the distributed algorithm of Yoshida (STOC 2011) for approximately solving the basic LP with techniques described in Saxena, Singer, Sudan, and Velusamy (SODA 2025) for simulating a distributed algorithm by a sublinear space streaming algorithm on bounded-degree instances of Max-DICUT, it appears that there are sublinear space streaming algorithms implementing the basic LP, for every CSP. Based on our results, we conjecture the following dichotomy theorem: Whenever the basic LP admits an integrality gap, there is a linear space single-pass streaming lower bound, and when the LP is roundable, there is a sublinear space streaming algorithm.

We construct a new family of explicit codes that are list decodable to capacity and achieve an optimal list size of O(1/epsilon). In contrast to existing explicit constructions of codes achieving list decoding capacity, our arguments do not rely on algebraic structure but utilize simple combinatorial properties of expander graphs. Our construction is based on a celebrated distance amplification procedure due to Alon, Edmonds, and Luby [FOCS95], which transforms any high-rate code into one with near-optimal rate-distance tradeoff. We generalize it to show that the same procedure can be used to transform any high-rate code into one that achieves list decoding capacity. Our proof can be interpreted as a local-to-global phenomenon for (a slight strengthening of) the generalized Singleton bound. Using this construction, for every R, epsilon is an element of (0,1) and k is an element of N+, we obtain an explicit family of rate R codes C subset of Sigma(n) that achieve the epsilon-relaxed generalized Singleton bound. The alphabet size of these codes is a constant depending only on epsilon and k, and they can be list decoded up to radius k-1/k center dot (1 - R - epsilon), in time n(k,epsilon)(O) (1) with a list of size k-1. As a corollary of our result, we also obtain the first explicit construction of LDPC codes achieving list decoding capacity, and in fact arbitrarily close to the generalized Singleton bound.

A set of high dimensional points X = {x(1), x(2), ... , x(n)} subset of R-d in isotropic position is said to be delta-anti concentrated if for every direction v, the fraction of points in X satisfying vertical bar < x(i), v >vertical bar <= delta is at most O(delta). Motivated by applications to list-decodable learning and clustering, three recent works [7], [44], [71] considered the problem of constructing efficient certificates of anti-concentration in the average case, when the set of points X corresponds to samples from a Gaussian distribution. Their certificates played a crucial role in several subsequent works in algorithmic robust statistics on list-decodable learning and settling the robust learnability of arbitrary Gaussian mixtures. Unlike related efficient certificates of concentration properties that are known for wide class of distributions [52], the aforementioned approach has been limited only to rotationally invariant distributions (and their affine transformations) with the only prominent example being Gaussian distributions. This work presents a new (and arguably the most natural) formulation for anti- concentration. Using this formulation, we give quasi-polynomial time verifiable sum-of-squares certificates of anti-concentration that hold for a wide class of non-Gaussian distributions including anti-concentrated bounded product distributions and uniform distributions over L-p balls (and their affine transformations). Consequently, our method upgrades and extends results in algorithmic robust statistics e.g., list-decodable learning and clustering, to such distributions. As in the case of previous works, our certificates are also obtained via relaxations in the sum-of-squares hierarchy. However, the nature of our argument differs significantly from prior works that formulate anti-concentration as the non-negativity of an explicit polynomial. Our argument constructs a canonical integer program for anti-concentration and analysis a SoS relaxation of it, independent of the intended application. The explicit polynomials appearing in prior works can be seen as specific dual certificates to this program. From a technical standpoint, unlike existing works that explicitly construct sum-of-squares certificates, our argument relies on duality and analyzes a pseudo-expectation on large subsets of the input points that take a small value in some direction. Our analysis uses the method of polynomial reweightings to reduce the problem to analyzing only analytically dense or sparse directions.

We give a construction of Quantum Low-Density Parity Check (QLDPC) codes with near-optimal rate-distance tradeoff and efficient list decoding up to the Johnson bound in polynomial time. Previous constructions of list decodable good distance quantum codes either required access to a classical side channel or were based on algebraic constructions that preclude the LDPC property. Our construction relies on new algorithmic results for codes obtained via the quantum analog of the distance amplification scheme of Alon, Edmonds, and Luby [FOCS 1995]. These results are based on convex relaxations obtained using the Sum-of-Squares hierarchy, which reduce the problem of list decoding the distance amplified codes to unique decoding the starting base codes. Choosing these base codes to be the recent breakthrough constructions of good QLDPC codes with efficient unique decoders, we get efficiently list decodable QLDPC codes.

We study the problem of computing the p \rightarrow q norm of a matrix A \in Rm\times n, defined as \| A\| p\rightarrow q = maxx\in Rn\setminus \{ 0\} \| x\| p. This problem generalizes the spectral norm of a matrix (p = q = 2) and the Grothendieck problem (p = \infty , q = 1) and has been widely studied in various regimes. When p \geq q, the problem exhibits a dichotomy: constant factor approximation algorithms are known if 2 \in [q, p], and the problem is hard to approximate within almost polynomial factors when 2 \in/ [q, p]. The regime when p < q, known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with p = 2 and q > 2 was studied by 307--326], who gave subexponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the exponential time hypothesis. However, no NP-hardness of approximation is known for these problems for any p < q. We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any 1 < p < q < \infty with 2 \in/ [p, q], \| A\| p\rightarrow q is hard to approximate within 2O((log n)1 - \epsilon ) assuming NP \subseteq \not BPTIME(2(logn)O(1) ). En route to the above result, we also prove almost tight results for the case when p \geq q with 2 \in [q, p].

We develop new list decoding algorithms for Tanner codes and distance-amplified codes based on bipartite spectral expanders. We show that proofs exhibiting lower bounds on the minimum distance of these codes can be used as certificates discoverable by relaxations in the Sum-of-Squares (SoS) semi-definite programming hierarchy. Combining these certificates with certain entropic proxies to ensure that the solutions to the relaxations cover the entire list, then leads to algorithms for list decoding several families of codes up to the Johnson bound. We prove the following results:- We show that the LDPC Tanner codes of Zémor [IEEE Trans. Inf. Theory 2001] with alphabet size q, block-length n and distance $\delta$, based on an expander graph with degree d, can be list-decoded up to distance $\mathcal{J}_{q}(\delta)-\varepsilon$ in time $n^{O_{d, q}\left(1 / \varepsilon^{4}\right)}$, where $\mathcal{J}_{q}(\delta)$ denotes the Johnson bound.- We show that the codes obtained via the expander-based distance amplification procedure of Alon, Edmonds and Luby [FOCS 1995] can be list-decoded close to the Johnson bound using the SoS hierarchy, by reducing the list decoding problem to unique decoding of the base code. In particular, starting from any base code unique-decodable up to distance $\delta$, one can obtain near-MDS codes with rate R and distance $1-R-\varepsilon$, list-decodable up to the Johnson bound in time $n^{O_{\varepsilon, \delta}(1)}$.- We show that the locally testable codes of Dinur et al. [STOC 2022] with alphabet size q, block-length n and distance $\delta$ based on a square Cayley complex with generator sets of size d, can be list-decoded up to distance $\mathcal{J}_{q}(\delta)-\varepsilon$ in time $n^{O_{d, q}\left(1 / \varepsilon^{4}\right)}$, where $\mathcal{J}_{q}(\delta)$ denotes the Johnson bound.

Analyzing concentration of large random matrices is a common task in a wide variety of fields. Given independent random variables, many tools are available to analyze random matrices whose entries are linear in the variables, e.g. the matrix-Bernstein inequality. However, in many applications, we need to analyze random matrices whose entries are polynomials in the variables. These arise naturally in the analysis of spectral algorithms, e.g., Hopkins et al. [STOC 2016], Moitra-Wein [STOC 2019]; and in lower bounds for semidefinite programs based on the Sum of Squares hierarchy, e.g. Barak et al. [FOCS 2016], Jones et al. [FOCS 2021]. In this work, we present a general framework to obtain such bounds, based on the matrix Efron-Stein inequalities developed by Paulin-Mackey-Tropp [Annals of Probability 2016]. The Efron-Stein inequality bounds the norm of a random matrix by the norm of another simpler (but still random) matrix, which we view as arising by "differentiating" the starting matrix. By recursively differentiating, our framework reduces the main task to analyzing far simpler matrices. For Rademacher variables, these simpler matrices are in fact deterministic and hence, analyzing them is far easier. For general non-Rademacher variables, the task reduces to scalar concentration, which is much easier. Moreover, in the setting of polynomial matrices, our results generalize the work of Paulin-Mackey-Tropp. Using our basic framework, we recover known bounds in the literature for simple "tensor networks" and "dense graph matrices". Using our general framework, we derive bounds for "sparse graph matrices", which were obtained only recently by Jones et al. [FOCS 2021] using a nontrivial application of the trace power method, and was a core component in their work. We expect our framework to be helpful for other applications involving concentration phenomena for nonlinear random matrices.

The ellipsoid fitting conjecture of Saunderson, Chandrasekaran, Parrilo and Willsky considers the maximum number $n$ random Gaussian points in $\mathbb{R}^d$, such that with high probability, there exists an origin-symmetric ellipsoid passing through all the points. They conjectured a threshold of $n = (1-o_d(1)) \cdot d^2/4$, while until recently, known lower bounds on the maximum possible $n$ were of the form $d^2/(\log d)^{O(1)}$. We give a simple proof based on concentration of sample covariance matrices, that with probability $1 - o_d(1)$, it is possible to fit an ellipsoid through $d^2/C$ random Gaussian points. Similar results were also obtained in two recent independent works by Hsieh, Kothari, Potechin and Xu [arXiv, July 2023] and by Bandeira, Maillard, Mendelson, and Paquette [arXiv, July 2023].

Abstract. We study the problem of computing the [Formula: see text] norm of a matrix [Formula: see text], defined as [Formula: see text]. This problem generalizes the spectral norm of a matrix ([Formula: see text]) and the Grothendieck problem ([Formula: see text], [Formula: see text]) and has been widely studied in various regimes. When [Formula: see text], the problem exhibits a dichotomy: constant factor approximation algorithms are known if [Formula: see text], and the problem is hard to approximate within almost polynomial factors when [Formula: see text]. The regime when [Formula: see text], known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with [Formula: see text] and [Formula: see text] was studied by Barak et al. [ Proceedings of the 44 th Annual ACM Symposium on Theory of Computing, 2012, pp. 307–326], who gave subexponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the exponential time hypothesis. However, no NP-hardness of approximation is known for these problems for any [Formula: see text]. We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any [Formula: see text] with [Formula: see text], [Formula: see text] is hard to approximate within [Formula: see text] assuming [Formula: see text]. En route to the above result, we also prove almost tight results for the case when [Formula: see text] with [Formula: see text].

For an abelian group $H$ acting on the set $[\ell]$, an $(H,\ell)$-lift of a graph $G_0$ is a graph obtained by replacing each vertex by $\ell$ copies, and each edge by a matching corresponding to the action of an element of $H$. In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group $H \leqslant \text{Sym}(\ell)$, constant degree $d \ge 3$ and $\epsilon > 0$, we construct explicit $d$-regular expander graphs $G$ obtained from an $(H,\ell)$-lift of a (suitable) base $n$-vertex expander $G_0$ with the following parameters: (i) $\lambda(G) \le 2\sqrt{d-1} + \epsilon$, for any lift size $\ell \le 2^{n^{\delta}}$ where $\delta=\delta(d,\epsilon)$, (ii) $\lambda(G) \le \epsilon \cdot d$, for any lift size $\ell \le 2^{n^{\delta_0}}$ for a fixed $\delta_0 > 0$, when $d \ge d_0(\epsilon)$, or (iii) $\lambda(G) \le \widetilde{O}(\sqrt{d})$, for lift size ``exactly'' $\ell = 2^{\Theta(n)}$. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes. Items $(i)$ and $(ii)$ above are obtained by extending the techniques of Mohanty, O'Donnell and Paredes [STOC 2020] for $2$-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing'" depth-first search traversals. Result $(iii)$ is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion.

The Sum-of-Squares (SoS) hierarchy of semidefinite programs is a powerful algorithmic paradigm which captures state-of-the-art algorithmic guarantees for a wide array of problems. In the average case setting, SoS lower bounds provide strong evidence of algorithmic hardness or information-computation gaps. Prior to this work, SoS lower bounds have been obtained for problems in the “dense” input regime, where the input is a collection of independent Rademacher or Gaussian random variables, while the sparse regime has remained out of reach. We make the first progress in this direction by obtaining strong SoS lower bounds for the problem of Independent Set on sparse random graphs. We prove that with high probability over an Erdós-Rénvi random graph $G\sim G_{n_{J}\frac{d}{u}}$ with average degree $d > \log^{2}n$ , degree-Dsos SoS fails to refute the existence of an independent set of size $k=\displaystyle \Omega(\frac{n}{\sqrt{d}(\log n)(\mathrm{D}_{\mathrm{S}\mathrm{o}\mathrm{S}})^{c_{0}}})$ in $G$ (where $c_{0}$ is an absolute constant), whereas the true size of the largest independent set in $G$ is $O(\displaystyle \frac{n\log d}{d})$ . Our proof involves several significant extensions of the techniques used for proving SoS lower bounds in the dense setting. Previous lower bounds are based on the pseudo-calibration heuristic of Barak et al. [FOCS 2016] which produces a candidate SoS solution using a planted distribution indistinguishable from the input distribution via low-degree tests. In the sparse case the natural planted distribution does admit low-degree distinguishers, and we show how to adapt the pseudo-calibration heuristic to overcome this. Another notorious technical challenge for the sparse regime is the quest for matrix norm bounds. In this paper, we obtain new norm bounds for graph matrices in the sparse setting. While in the dense setting the norms of graph matrices are characterized by the size of the minimum vertex separator of the corresponding graph, this turns not to be the case for sparse graph matrices. Another contribution of our work is developing a new combinatorial understanding of structures needed to understand the norms of sparse graph matrices.

Grothendieck’s inequality [Gro53] states that there is an absolute constant K > 1 such that for any n × n matrix A

The Gilbert–Varshamov bound non-constructively establishes the existence of binary codes of distance 1/2−є/2 and rate Ω(є 2 ). In a breakthrough result, Ta-Shma [STOC 2017] constructed the first explicit family of nearly optimal binary codes with distance 1/2−є/2 and rate Ω(є 2+α ), where α → 0 as є → 0. Moreover, the codes in Ta-Shma’s construction are є-balanced, where the distance between distinct codewords is not only bounded from below by 1/2−є/2, but also from above by 1/2+є/2. Polynomial time decoding algorithms for (a slight modification of) Ta-Shma’s codes appeared in [FOCS 2020], and were based on the Sum-of-Squares (SoS) semidefinite programming hierarchy. The running times for these algorithms were of the form N O α (1) for unique decoding, and N O є,α (1) for the setting of “gentle list decoding”, with large exponents of N even when α is a fixed constant. We derive new algorithms for both these tasks, running in time Õ є ( N ). Our algorithms also apply to the general setting of decoding direct-sum codes. Our algorithms follow from new structural and algorithmic results for collections of k -tuples (ordered hypergraphs) possessing a “structured expansion” property, which we call splittability . This property was previously identified and used in the analysis of SoS-based decoding and constraint satisfaction algorithms, and is also known to be satisfied by Ta-Shma’s code construction. We obtain a new weak regularity decomposition for (possibly sparse) splittable collections W ⊆ [ n ] k , similar to the regularity decomposition for dense structures by Frieze and Kannan [FOCS 1996]. These decompositions are also computable in near-linear time Õ(| W |), and form a key component of our algorithmic results.

Using the previous discussion, we can write matrices in convenient form. Let A ∈ Cm×n, which can be thought of as an operator from Cn to Cm. Let σ1, . . . , σr be the non-zero singular values and let v1, . . . , vr and w1, . . . , wr be the right and left singular vectors respectively. Note that V = Cn and W = Cm and v ∈ V, w ∈ W, we can write the operator |w〉 〈v| as the matrix wv∗, where v∗ denotes vT. This is because for any u ∈ V, wv∗u = w(v∗u) = 〈v, u〉 · w. Thus, we can write

Proof: Complete u1, . . . , uk to an orthonormal basis uk+1, . . . , ud for all of Rd. For any point v ∈ Rd, where exist c1, . . . , cd ∈ R such that v = ∑j=1 cj · uj. To find the distance dist(v, S) = minu∈S ∥v − u∥, we need to find the point u ∈ S, which is closest to v. Let u = ∑j=1 bj · uk be an arbitrary point in S (any u ∈ S can be written in this form, since u1, . . . , uk form a basis for S). We have that

1 Solving systems of linear equations: Gaussian elimination Given a system of linear equations Ax = b for A ∈ Fm×n, b ∈ Fm, recall that we can solve the system or determine that there is no solution by converting the matrix [A | b] to a row-reduced form using elementary row operations. Definition 1.1 A matrix M ∈ Fm×n is said to be in row-reduced form if The first non-zero entry in each row (known as the leading entry) is 1. If the leading entry in row i0 is in column j0, then Mij = 0 for all i > i0 and j ≤ j0. All non-zero rows occur above the zero rows. Notice that a matrix in the row-reduced form is always upper triangular. The system has no solution if and only if there is a non-zero row with a leading entry in the last column (corresponding to the entries of b). Also, if the system has a solution, then it can easily be found using back-substitution, starting from the last non-zero row. Also, recall that an elementary row operations consist of the following (using Mi to denote the ith row of M): Swapping the rows Mi and Mj, for some i, j,∈ [m]. Mi ← c ·Mi for some i ∈ [m], c ∈ F \ {0}. Mi ← Mi + c ·Mj for some i, j ∈ [m], c ∈ F. A matrix M can always be converted to a row-reduced form using elementary row operations, which gives a general algorithm for solving a system of linear equations over any field. However, the time taken by this algorithm can be as large as Ω(n3), which is prohibitive for large matrices. In the next lecture, we will discuss methods which can take advantage of sparsity to significantly speed up the solution of linear systems.