In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a polyLaplacian regulariser. The methodology is readily adapted to graphs and here we consider graph polyLaplacian regularization in a fully supervised, nonparametric, noise corrupted, regression problem. In particular, given a dataset
Let
 NSFPAR ID:
 10361768
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 The Journal of Geometric Analysis
 Volume:
 32
 Issue:
 2
 ISSN:
 10506926
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract and a set of noisy labels$$\{x_i\}_{i=1}^n$$ ${\left\{{x}_{i}\right\}}_{i=1}^{n}$ we let$$\{y_i\}_{i=1}^n\subset \mathbb {R}$$ ${\left\{{y}_{i}\right\}}_{i=1}^{n}\subset R$ be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph polyLaplacian term. When$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$ ${u}_{n}:{\left\{{x}_{i}\right\}}_{i=1}^{n}\to R$ , for iid noise$$y_i = g(x_i)+\xi _i$$ ${y}_{i}=g\left({x}_{i}\right)+{\xi}_{i}$ , and using the geometric random graph, we identify (with high probability) the rate of convergence of$$\xi _i$$ ${\xi}_{i}$ to$$u_n$$ ${u}_{n}$g in the large data limit . Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.$$n\rightarrow \infty $$ $n\to \infty $ 
Abstract Let us fix a prime
p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ ${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$ with coefficients$$j=1,\dots ,m$$ $j=1,\cdots ,m$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ ${a}_{j,i}\in {F}_{p}$ , that$$k\ge 3m$$ $k\ge 3m$ for$$a_{j,1}+\dots +a_{j,k}=0$$ ${a}_{j,1}+\cdots +{a}_{j,k}=0$ and that every$$j=1,\dots ,m$$ $j=1,\cdots ,m$ minor of the$$m\times m$$ $m\times m$ matrix$$m\times k$$ $m\times k$ is nonsingular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ ${\left({a}_{j,i}\right)}_{j,i}$n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq {F}_{p}^{n}$ contains a solution$$A> C\cdot \Gamma ^n$$ $\leftA\right>C\xb7{\Gamma}^{n}$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ $({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ ${x}_{1},\cdots ,{x}_{k}\in A$C and are constants only depending on$$\Gamma $$ $\Gamma $p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma $\Gamma <p$
in the solution$$x_1,\dots ,x_k$$ ${x}_{1},\cdots ,{x}_{k}$ to be distinct. If we relax this condition and only demand that$$(x_1,\dots ,x_k)\in A^k$$ $({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a nondiagonal tensor in combination with combinatorial and probabilistic arguments.$$x_1,\dots ,x_k$$ ${x}_{1},\cdots ,{x}_{k}$ 
Abstract Approximate integer programming is the following: For a given convex body
, either determine whether$$K \subseteq {\mathbb {R}}^n$$ $K\subseteq {R}^{n}$ is empty, or find an integer point in the convex body$$K \cap {\mathbb {Z}}^n$$ $K\cap {Z}^{n}$ which is$$2\cdot (K  c) +c$$ $2\xb7(Kc)+c$K , scaled by 2 from its center of gravityc . Approximate integer programming can be solved in time while the fastest known methods for exact integer programming run in time$$2^{O(n)}$$ ${2}^{O\left(n\right)}$ . So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$2^{O(n)} \cdot n^n$$ ${2}^{O\left(n\right)}\xb7{n}^{n}$ can be found in time$$x^* \in (K \cap {\mathbb {Z}}^n)$$ ${x}^{\ast}\in (K\cap {Z}^{n})$ , provided that the$$2^{O(n)}$$ ${2}^{O\left(n\right)}$remainders of each component for some arbitrarily fixed$$x_i^* \mod \ell $$ ${x}_{i}^{\ast}\phantom{\rule{0ex}{0ex}}mod\phantom{\rule{0ex}{0ex}}\ell $ of$$\ell \ge 5(n+1)$$ $\ell \ge 5(n+1)$ are given. The algorithm is based on a$$x^*$$ ${x}^{\ast}$cuttingplane technique , iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a new$$2^{O(n)}n^n$$ ${2}^{O\left(n\right)}{n}^{n}$asymmetric approximate Carathéodory theorem that might be of interest on its own. Our second method concerns integer programming problems in equationstandard form . Such a problem can be reduced to the solution of$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$ $Ax=b,0\le x\le u,\phantom{\rule{0ex}{0ex}}x\in {Z}^{n}$ approximate integer programming problems. This implies, for example that$$\prod _i O(\log u_i +1)$$ ${\prod}_{i}O(log{u}_{i}+1)$knapsack orsubsetsum problems withpolynomial variable range can be solved in time$$0 \le x_i \le p(n)$$ $0\le {x}_{i}\le p\left(n\right)$ . For these problems, the best running time so far was$$(\log n)^{O(n)}$$ ${(logn)}^{O\left(n\right)}$ .$$n^n \cdot 2^{O(n)}$$ ${n}^{n}\xb7{2}^{O\left(n\right)}$ 
Abstract We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ $\mathrm{Quasi}\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$ , by showing that$\mathcal { C}$ $C$ admits nontrivial satisfiability and/or$\mathcal { C}$ $C$# SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a nontrivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ $C$f (x _{1},…,x _{n}) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ ${\sum}_{i}{x}_{i}$f , and for all “typical” , faster$\mathcal { C}$ $C$# SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ $C$ , which may be$f \circ \mathcal { C}$ $f\circ C$stronger than itself. In particular:$\mathcal { C}$ $C$# SAT algorithms forn ^{k}size circuits running in 2^{n}/$\mathcal { C}$ $C$n ^{k}time (for allk ) implyN E X P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$# SAT algorithms for size$2^{n^{{\varepsilon }}}$ ${2}^{{n}^{\epsilon}}$ circuits running in$\mathcal { C}$ $C$ time (for some$2^{nn^{{\varepsilon }}}$ ${2}^{n{n}^{\epsilon}}$ε > 0) implyQ u a s i N P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i N P does not haveE M A J ∘A C C ^{0}∘T H R circuits of polynomial size, whereE M A J is the “exact majority” function, improving previous lower bounds againstA C C ^{0}[Williams JACM’14] andA C C ^{0}∘T H R [Williams STOC’14], [MurrayWilliams STOC’18]. This is the first nontrivial lower bound against such a circuit class. 
Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to {R}_{+}$r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ ${f}_{1},{f}_{2},\dots ,{f}_{r}$N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ ${k}_{1},{k}_{2},\dots ,{k}_{r}$ such that$$S \subseteq N$$ $S\subseteq N$ for$$f_i(S) \ge k_i$$ ${f}_{i}\left(S\right)\ge {k}_{i}$ . We refer to this problem as$$1 \le i \le r$$ $1\le i\le r$MultiSubmodCover and it was recently considered by HarPeled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 HarPeled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$MultiSubmodCover generalizes the wellknown Submodular Set Cover problem (SubmodSC ), and it can also be easily reduced toSubmodSC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ $O(log(kr\left)\right)$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for$$k = \sum _i k_i$$ $k={\sum}_{i}{k}_{i}$MultiSubmodCover that covers each constraint to within a factor of while incurring an approximation of$$(11/e\varepsilon )$$ $(11/e\epsilon )$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{\u03f5}logr)$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ ${f}_{i}$PartialSC ), covering integer programs (CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the highlevel model and the lens of submodularity in addressing this class of covering problems.