Computational Hardness of Certifying Bounds on Constrained PCA Problems

Given a random n×n symmetric matrix W drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form x⊤Wx over all vectors x in a constraint set S⊂Rn. For a certain class of normalized constraint sets S we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of W. A notable special case included in our results is the hypercube S={±1/n−−√}n, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics. Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over x∈S is much larger than that of a GOE matrix.ISSN:1868-896

Rounding Sum-of-Squares Relaxations

We present a general approach to rounding semidefinite programming relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our approach is based on using the connection between these relaxations and the Sum-of-Squares proof system to transform a *combining algorithm* -- an algorithm that maps a distribution over solutions into a (possibly weaker) solution -- into a *rounding algorithm* that maps a solution of the relaxation to a solution of the original problem. Using this approach, we obtain algorithms that yield improved results for natural variants of three well-known problems: 1) We give a quasipolynomial-time algorithm that approximates the maximum of a low degree multivariate polynomial with non-negative coefficients over the Euclidean unit sphere. Beyond being of interest in its own right, this is related to an open question in quantum information theory, and our techniques have already led to improved results in this area (Brand\~{a}o and Harrow, STOC '13). 2) We give a polynomial-time algorithm that, given a d dimensional subspace of R^n that (almost) contains the characteristic function of a set of size n/k, finds a vector $v$ in the subspace satisfying $|v|_4^4 > c(k/d^{1/3}) |v|_2^2$, where $|v|_p = (E_i v_i^p)^{1/p}$. Aside from being a natural relaxation, this is also motivated by a connection to the Small Set Expansion problem shown by Barak et al. (STOC 2012) and our results yield a certain improvement for that problem. 3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time algorithm with substantially improved guarantees for recovering a planted $\mu$-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n nonzero coordinates, we can recover it with high probability whenever $\mu < O(\min(1,n/d^2))$, improving for $d < n^{2/3}$ prior methods which intrinsically required $\mu < O(1/\sqrt(d))$

Heterosis Studies for Some Morphological, Seed Yield and Quality Traits in Rapeseed (Brassica napus L.)

Heterosis has a significant position in rapeseed breeding. To assess the heterosis for seed yield and quality traits, three Brassica napus L. testers and five lines were crossed using line × tester design in RCBD with three replications to obtain cross seeds of fifteen hybrids. Data of fifteen characters were recorded. Mean sum of squares of analysis of variances for genotypes were significant or highly significant for all of the fifteen traits. Low to High degree of desirable heterosis over mid, better and commercial parents were observed. Cross 13 showed maximum values of siliqua length (14.3%, 11.1%), seed yield/plant (45.3%, 35.9%) and LnicC (-43.7%, -37.6%) for MPH and BPH as well as LnicC (-38.3%) for CH. Cross 3 revealed highest PC (5.5%, 4.4%), Cross 4 for NSP (28.4%, 25.3%), Cross 10 for GLC (-13.5%, -33.2%) and Cross 15 for NSS (22.8%, 10.8%) over MPH and BPH. Maximum OC (9.3%, 6.9%) was revealed by Cross 8 for BPH and CH. Cross No. 1 possessed highest heterosis over commercial variety ‘Punjab Sarson’ for PC (21.2%), OAC (10.8%), LeicC (46.8%), DM (-6.8%), EAC (-36.9%) and GLC (-29.3%). Cross 6 revealed maximum CH for SY (73.3%) and DF (-10.8%). The present study provides valuable facts of noble hybrids with improved traits related to nutrition and yield, as well as valuable information for further molecular and genetic studies of heterosis for these agronomic traits in B. napus. Keywords: Brassica napus L., Line × Tester, Heterosis, Morphological, Seed qualit

The power of sum-of-squares for detecting hidden structures

We study planted problems---finding hidden structures in random noisy inputs---through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of accuracy of recovered solutions and robustness to noise. One theme in recent work is the design of spectral algorithms which match the guarantees of SoS algorithms for planted problems. Classical spectral algorithms are often unable to accomplish this: the twist in these new spectral algorithms is the use of spectral structure of matrices whose entries are low-degree polynomials of the input variables. We prove that for a wide class of planted problems, including refuting random constraint satisfaction problems, tensor and sparse PCA, densest-k-subgraph, community detection in stochastic block models, planted clique, and others, eigenvalues of degree-d matrix polynomials are as powerful as SoS semidefinite programs of roughly degree d. For such problems it is therefore always possible to match the guarantees of SoS without solving a large semidefinite program. Using related ideas on SoS algorithms and low-degree matrix polynomials (and inspired by recent work on SoS and the planted clique problem by Barak et al.), we prove new nearly-tight SoS lower bounds for the tensor and sparse principal component analysis problems. Our lower bounds for sparse principal component analysis are the first to suggest that going beyond existing algorithms for this problem may require sub-exponential time

Lower bounds on the size of semidefinite programming relaxations

We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on $n$-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than $2^{n^c}$, for some constant $c > 0$. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-$O(1)$ sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for MAX-3-SAT

Study of low flow rate ladle bottom gas stirring using triaxial vibration signals

Secondary steelmaking plays a great role in enhancing the quality of the final steel product. The metal quality is a function of metal bath stirring in ladles. The metal bath is often stirred by an inert gas to achieve maximum compositional and thermal uniformity throughout the melt. Ladle operators often observe the top surface phenomena, such as level of meniscus disturbance, to evaluate the status of stirring. However, this type of monitoring has significant limitations in assessing the process accurately especially at low gas flow rate bubbling. The present study investigates stirring phenomena using ladle wall triaxial vibration at a low flow rate on a steel-made laboratory model and plant scale for the case of the vacuum tank degasser. Cold model and plant data were successfully modeled by partial least-squares regression to predict the amount of stirring. In the cold model, it was found that the combined vibration signal could predict the stirring power and recirculation speed effectively in specific frequency ranges. Plant trials also revealed that there is a high structure in each data set and in the same frequency ranges at the water model. In the case of industrial data, the degree of linear relationship was strong for data taken from a single heat