Persistent homology is a popular method for computing topological features of (metric) data. Standard approaches based on the Čech or Rips filtration are stable under small perturbations of the data, but highly sensitive to outliers. This lack of robustness has been frequently addressed in the literature. In this paper, we take a novel perspective by asking the following question: When can we guarantee that an observed persistent feature (a bar) is inherent to the underlying data in the presence of a limited number of unknown, arbitrary outliers. We formalize this question by introducing the notion of adversarial robustness, and study the problem of deciding whether a given bar in the barcode of a filtered simplicial complex is adversarially robust. We show that this problem is essentially equivalent to a homological variant of the minimum cut problem in simplicial complexes, which we believe to be of independent interest. As our main technical contribution, we provide the first computational complexity results for this problem, consisting of an efficient algorithm in 0-dimensional homology, NP-hardness for the general problem, and an efficient algorithm for codimension-1 in n-dimensional complexes embedded in ℝⁿ. We also analyze its natural linear programming relaxation, whose dual defines a homological analog of the max-flow problem in graphs. We show that a max-flow/min-cut theorem does not hold in our setting, implying that the LP relaxation is not tight in general. Finally, in the special case of the Rips filtration, we provide a global heuristic based on the Hausdorff distance that guarantees adversarial robustness of sufficiently long bars. This connects adversarial robustness to standard stability theorems in persistent homology.

We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of \(\sum _{i=1}^n x_i\) for 0/1 variables under the constraint \(\sum _{i=1}^n x_i \ge q\), with \(q \in \mathbb {R}\). Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász–Schrijver and Sherali–Adams hierarchies require (maximal) rank n to solve them, already when q = 1/2 is constant. The SOS hierarchy requires only sublinear rank \(O(\sqrt {n})\) to solve unweighted MK when q = 1/2. On the other hand, when q is allowed to vary with n, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when q is large, and when q is very small (0 < q ≤ 2− n). This raises the question of whether we should think of hard instances of unweighted MK as being typical for the SOS hierarchy, or as a consequence of very specific choices of the threshold parameter q.

Efficient algorithms for convex optimization, such as the ellipsoid method, require an a priori bound on the radius of a ball around the origin guaranteed to contain an optimal solution if one exists. For linear and convex quadratic programming, such solution bounds follow from classical characterizations of optimal solutions by systems of linear equations. For other programs, e.g., semidefinite ones, examples due to Khachiyan show that optimal solutions may require huge coefficients with an exponential number of bits, even if we allow approximations. Correspondingly, semidefinite programming is not even known to be in NP.

We investigate implications of the (extended) low-degree conjecture (recently formalized in [moitra et al2023]) in the context of the symmetric stochastic block model. Assuming the conjecture holds, we establish that no polynomial-time algorithm can weakly recover community labels below the Kesten-Stigum (KS) threshold. In particular, we rule out polynomial-time estimators that, with constant probability, achieve $n^{-0.49}$ correlation with the true communities. Whereas, above the KS threshold, polynomial-time algorithms are known to achieve constant correlation with the true communities with high probability [massoulie et al 2014,abbe et al 2015]. To our knowledge, we provide the first rigorous evidence for such sharp transition in recovery rate for polynomial-time algorithms at the KS threshold. Notably, under a stronger version of the low-degree conjecture, our lower bound remains valid even when the number of blocks diverges. Furthermore, our results provide evidence of a computational-to-statistical gap in learning the parameters of stochastic block models. In contrast, prior work either (i) rules out polynomial-time algorithms with $1 - o(1)$ success probability [Hopkins 18, bandeira et al 2021] under the low-degree conjecture, or (ii) degree-$\text{poly}(k)$ polynomials for learning the stochastic block model [Luo et al 2023]. For this, we design a hypothesis test which succeeeds with constant probability under symmetric stochastic block model, and $1-o(1)$ probability under the distribution of \Erdos \Renyi random graphs. Our proof combines low-degree lower bounds from [Hopkins 18, bandeira et al 2021] with graph splitting and cross-validation techniques. In order to rule out general recovery algorithms, we employ the correlation preserving projection method developed in [Hopkins et al 17].

Rubinfeld \& Vasilyan recently introduced the framework of *testable learning* as an extension of the classical agnostic model. It relaxes distributional assumptions which are difficult to verify by conditions that can be checked efficiently by a *tester*. The tester has to accept whenever the data truly satisfies the original assumptions, and the learner has to succeed whenever the tester accepts. We focus on the setting where the tester has to accept standard Gaussian data. There, it is known that basic concept classes such as halfspaces can be learned testably with the same time complexity as in the (distribution-specific) agnostic model. In this work, we ask whether there is a price to pay for testably learning more complex concept classes. In particular, we consider polynomial threshold functions (PTFs), which naturally generalize halfspaces. We show that PTFs of arbitrary constant degree can be testably learned up to excess error $\varepsilon > 0$ in time $n^{\mathrm{poly}(1/\varepsilon)}$. This qualitatively matches the best known guarantees in the agnostic model. Our results build on a connection between testable learning and *fooling*. In particular, we show that distributions that approximately match at least $\mathrm{poly}(1/\varepsilon)$ moments of the standard Gaussian fool constant-degree PTFs (up to error $\varepsilon$). As a secondary result, we prove that a direct approach to show testable learning (without fooling), which was successfully used for halfspaces, cannot work for PTFs.

Persistent homology has been widely used to study the topology of point clouds in Rn. Standard approaches are very sensitive to outliers, and their computational complexity depends badly on the number of data points. In this paper we introduce a novel persistence module for a point cloud using the theory of Christoffel-Darboux kernels. This module is robust to (statistical) outliers in the data, and can be computed in time linear in the number of data points. We illustrate the benefits and limitations of our new module with various numerical examples in Rn, for n = 1, 2, 3. Our work expands upon recent applications of Christoffel-Darboux kernels in the context of statistical data analysis and geometric inference [13]. There, these kernels are used to construct a polynomial whose level sets capture the geometry of a point cloud in a precise sense. We show that the persistent homology associated to the sublevel set filtration of this polynomial is stable with respect to the Wasserstein distance. Moreover, we show that the persistent homology of this filtration can be computed in singly exponential time in the ambient dimension n, using a recent algorithm of Basu & Karisani [1].

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O’Donnell [16] and later Raghavendra & Weitz [20] show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz [20]. In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.