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Depth-3 circuit lower bounds and $k$-SAT algorithms are intimately related; the state-of-the-art $\Sigma^k_3$-circuit lower bound and the $k$-SAT algorithm are based on the same combinatorial theorem. In this paper we define a problem which reveals new interactions between the two. Define Enum($k$, $t$) problem as: given an $n$-variable $k$-CNF and an initial assignment $\alpha$, output all satisfying assignments at Hamming distance $t$ from $\alpha$, assuming that there are no satisfying assignments of Hamming distance less than $t$ from $\alpha$. Observe that: an upper bound $b(n, k, t)$ on the complexity of Enum($k$, $t$) implies: - Depth-3 circuits: Any $\Sigma^k_3$ circuit computing the Majority function has size at least $\binom{n}{\frac{n}{2}}/b(n, k, \frac{n}{2})$. - $k$-SAT: There exists an algorithm solving $k$-SAT in time $O(\sum_{t = 1}^{n/2}b(n, k, t))$. A simple construction shows that $b(n, k, \frac{n}{2}) \ge 2^{(1 - O(\log(k)/k))n}$. Thus, matching upper bounds would imply a $\Sigma^k_3$-circuit lower bound of $2^{\Omega(\log(k)n/k)}$ and a $k$-SAT upper bound of $2^{(1 - \Omega(\log(k)/k))n}$. The former yields an unrestricted depth-3 lower bound of $2^{\omega(\sqrt{n})}$ solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum($k$, $t$) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum($3$, $\frac{n}{2}$). We show that the expected running time of our algorithm is $1.598^n$, substantially improving on the trivial bound of $3^{n/2} \simeq 1.732^n$. This already improves $\Sigma^3_3$ lower bounds for Majority function to $1.251^n$. The previous bound was $1.154^n$ which follows from the work of H{\aa}stad, Jukna, and Pudl\'ak (Comput. Complex.'95).

In dimension $d$, Mutually Unbiased Bases (MUBs) are a collection of orthonormal bases over $\mathbb{C}^d$ such that for any two vectors $v_1, v_2$ belonging to different bases, the scalar product $|\braket{v_1|v_2}| = \frac{1}{\sqrt{d}}$. The upper bound on the number of such bases is $d+1$. Constructions to achieve this bound are known when $d$ is some power of prime. The situation is more restrictive in other cases and also when we consider the results over real rather than complex. Thus, certain relaxations of this model are considered in literature and consequently Approximate MUBs (AMUB) are studied. This enables one to construct potentially large number of such objects for $\mathbb{C}^d$ as well as in $\mathbb{R}^d$. In this regard, we propose the concept of Almost Perfect MUBs (APMUB), where we restrict the absolute value of inner product $|\braket{v_1|v_2}|$ to be two-valued, one being 0 and the other $ \leq \frac{1+\mathcal{O}(d^{-\lambda})}{\sqrt{d}}$, such that $\lambda > 0$ and the numerator $1 + \mathcal{O}(d^{-\lambda}) \leq 2$. Each such vector constructed, has an important feature that large number of its components are zero and the non-zero components are of equal magnitude. Our techniques are based on combinatorial structures related to RBDs. We show that for several composite dimensions $d$, one can construct $\mathcal{O}(\sqrt{d})$ many APMUBs, in which cases the number of MUBs are significantly small. To be specific, this result works for $d$ of the form $(q-e)(q+f), \ q, e, f \in \mathbb{N}$, with the conditions $0 \leq f \leq e$ for constant $e, f$ and $q$ some power of prime. We also show that such APMUBs provide sets of Bi-angular vectors which are $\mathcal{O}(d^{\frac{3}{2}})$ in numbers, having high angular distances among them. Finally, as the MUBs are equivalent to a set of Hadamard matrices, we show that the APMUBs are so with the set of Weighing matrices.

We give a procedure for computing group-level $(\epsilon, \delta)$-DP guarantees for DP-SGD, when using Poisson sampling or fixed batch size sampling. Up to discretization errors in the implementation, the DP guarantees computed by this procedure are tight (assuming we release every intermediate iterate).

Recently (Elkin, Filtser, Neiman 2017) introduced the concept of a {\it terminal embedding} from one metric space $(X,d_X)$ to another $(Y,d_Y)$ with a set of designated terminals $T\subset X$. Such an embedding $f$ is said to have distortion $\rho\ge 1$ if $\rho$ is the smallest value such that there exists a constant $C>0$ satisfying \begin{equation*} \forall x\in T\ \forall q\in X,\ C d_X(x, q) \le d_Y(f(x), f(q)) \le C \rho d_X(x, q) . \end{equation*} When $X,Y$ are both Euclidean metrics with $Y$ being $m$-dimensional, recently (Narayanan, Nelson 2019), following work of (Mahabadi, Makarychev, Makarychev, Razenshteyn 2018), showed that distortion $1+\epsilon$ is achievable via such a terminal embedding with $m = O(\epsilon^{-2}\log n)$ for $n := |T|$. This generalizes the Johnson-Lindenstrauss lemma, which only preserves distances within $T$ and not to $T$ from the rest of space. The downside of prior work is that evaluating their embedding on some $q\in \mathbb{R}^d$ required solving a semidefinite program with $\Theta(n)$ constraints in~$m$ variables and thus required some superlinear $\mathrm{poly}(n)$ runtime. Our main contribution in this work is to give a new data structure for computing terminal embeddings. We show how to pre-process $T$ to obtain an almost linear-space data structure that supports computing the terminal embedding image of any $q\in\mathbb{R}^d$ in sublinear time $O^* (n^{1-\Theta(\epsilon^2)} + d)$. To accomplish this, we leverage tools developed in the context of approximate nearest neighbor search.

A subset $S$ of vertices in a graph $G$ is a secure dominating set of $G$ if $S$ is a dominating set of $G$ and, for each vertex $u \not\in S$, there is a vertex $v \in S$ such that $uv$ is an edge and $(S \setminus \{v\}) \cup \{u\}$ is also a dominating set of $G$. The secure domination number of $G$, denoted by $\gamma_{s}(G)$, is the cardinality of a smallest secure dominating sets of $G$. In this paper, we prove that for any outerplanar graph with $n \geq 4$ vertices, $\gamma_{s}(G) \geq (n+4)/5$ and the bound is tight.

For a $P$-indexed persistence module ${\sf M}$, the (generalized) rank of ${\sf M}$ is defined as the rank of the limit-to-colimit map for ${\sf M}$ over the poset $P$. For $2$-parameter persistence modules, recently a zigzag persistence based algorithm has been proposed that takes advantage of the fact that generalized rank for $2$-parameter modules is equal to the number of full intervals in a zigzag module defined on the boundary of the poset. Analogous definition of boundary for $d$-parameter persistence modules or general $P$-indexed persistence modules does not seem plausible. To overcome this difficulty, we first unfold a given $P$-indexed module ${\sf M}$ into a zigzag module ${\sf M}_{ZZ}$ and then check how many full interval modules in a decomposition of ${\sf M}_{ZZ}$ can be folded back to remain full in ${\sf M}$. This number determines the generalized rank of ${\sf M}$. For special cases of degree-$d$ homology for $d$-complexes, we obtain a more efficient algorithm including a linear time algorithm for degree-$1$ homology in graphs.

Let $X$ and $Z$ be random vectors, and $Y=g(X,Z)$. In this paper, on the one hand, for the case that $X$ and $Z$ are continuous, by using the ideas from the total variation and the flux of $g$, we develop a point of view in causal inference capable of dealing with a broad domain of causal problems. Indeed, we focus on a function, called Probabilistic Easy Variational Causal Effect (PEACE), which can measure the direct causal effect of $X$ on $Y$ with respect to continuously and interventionally changing the values of $X$ while keeping the value of $Z$ constant. PEACE is a function of $d\ge 0$, which is a degree managing the strengths of probability density values $f(x|z)$. On the other hand, we generalize the above idea for the discrete case and show its compatibility with the continuous case. Further, we investigate some properties of PEACE using measure theoretical concepts. Furthermore, we provide some identifiability criteria and several examples showing the generic capability of PEACE. We note that PEACE can deal with the causal problems for which micro-level or just macro-level changes in the value of the input variables are important. Finally, PEACE is stable under small changes in $\partial g_{in}/\partial x$ and the joint distribution of $X$ and $Z$, where $g_{in}$ is obtained from $g$ by removing all functional relationships defining $X$ and $Z$.

Conformalized Quantile Regression (CQR) is a recently proposed method for constructing prediction intervals for a response $Y$ given covariates $X$, without making distributional assumptions. However, existing constructions of CQR can be ineffective for problems where the quantile regressors perform better in certain parts of the feature space than others. The reason is that the prediction intervals of CQR do not distinguish between two forms of uncertainty: first, the variability of the conditional distribution of $Y$ given $X$ (i.e., aleatoric uncertainty), and second, our uncertainty in estimating this conditional distribution (i.e., epistemic uncertainty). This can lead to intervals that are overly narrow in regions where epistemic uncertainty is high. To address this, we propose a new variant of the CQR methodology, Uncertainty-Aware CQR (UACQR), that explicitly separates these two sources of uncertainty to adjust quantile regressors differentially across the feature space. Compared to CQR, our methods enjoy the same distribution-free theoretical coverage guarantees, while demonstrating in our experiments stronger conditional coverage properties in simulated settings and real-world data sets alike.

The large-matrix limit laws of the rescaled largest eigenvalue of the orthogonal, unitary and symplectic $n$-dimensional Gaussian ensembles -- and of the corresponding Laguerre ensembles (Wishart distributions) for various regimes of the parameter $\alpha$ (degrees of freedom $p$) -- are known to be the Tracy-Widom distributions $F_\beta$ ($\beta=1,2,4$). We will establish (paying particular attention to large, or small, ratios $p/n$) that, with careful choices of the rescaling constants and the expansion parameter $h$, the limit laws embed into asymptotic expansions in powers of $h$, where $h \asymp n^{-2/3}$ resp. $h \asymp (n\,\wedge\,p)^{-2/3}$. We find explicit analytic expressions of the first few expansions terms as linear combinations, with rational polynomial coefficients, of higher order derivatives of the limit law $F_\beta$. With a proper parametrization, the expansions in the Gaussian cases can be understood, for given $n$, as the limit $p\to\infty$ of the Laguerre cases. Whereas the results for $\beta=2$ are presented with proof, the discussion of the cases $\beta=1,4$ is based on some hypotheses, focussing on the algebraic aspects of actually computing the polynomial coefficients. For the purposes of illustration and validation, the various results are checked against simulation data with a sample size of a thousand million.

Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability $p \in (0,1/2)$. As our main result, we show that, when $k = o(n)$, it is both sufficient and necessary to use $$(1 \pm o(1)) \frac{n\log \frac{k}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$$ queries in expectation to compute the $\mathsf{TH}_k$ function with a vanishing error probability $\delta = o(1)$, where $D_{\mathsf{KL}}(p || 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. In particular, this says that $(1 \pm o(1)) \frac{n\log \frac{1}{\delta}}{D_{\mathsf{KL}}(p || 1-p)}$ queries in expectation are both sufficient and necessary to compute the $\mathsf{OR}$ and $\mathsf{AND}$ functions of $n$ Boolean variables. Compared to previous work, our result tightens the dependence on $p$ in both the upper and lower bounds.