Semi-random Model Research Articles

Rank-One Boolean Tensor Factorization and the Multilinear Polytope

We consider the NP-hard problem of finding the closest rank-one binary tensor to a given binary tensor, which we refer to as the rank-one Boolean tensor factorization (BTF) problem. This optimization problem can be used to recover a planted rank-one tensor from noisy observations. We formulate rank-one BTF as the problem of minimizing a linear function over a highly structured multilinear set. Leveraging on our prior results regarding the facial structure of multilinear polytopes, we propose novel linear programming relaxations for rank-one BTF. We then establish deterministic sufficient conditions under which our proposed linear programs recover a planted rank-one tensor. To analyze the effectiveness of these deterministic conditions, we consider a semirandom model for the noisy tensor and obtain high probability recovery guarantees for the linear programs. Our theoretical results as well as numerical simulations indicate that certain facets of the multilinear polytope significantly improve the recovery properties of linear programming relaxations for rank-one BTF. Funding: A. Del Pia is partially funded by the Air Force Office of Scientific Research [Grant FA9550-23-1-0433]. A. Khajavirad is partially funded by the Air Force Office of Scientific Research [Grant FA9550-23-1-0123].

Hidden Integrality and Semirandom Robustness of SDP Relaxation for Sub-Gaussian Mixture Model

We consider the problem of estimating the discrete clustering structures under the sub-Gaussian mixture model. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solution to the SDP is not integer-valued in general, its estimation error can be upper bounded by that of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. In addition, we show that the SDP relaxation is robust under the semirandom setting in which an adversary can modify the data generated from the mixture model. In particular, we generalize the hidden integrality property to the semirandom model and thereby show that SDP achieves the optimal error bound in this setting. These results together highlight the “global-to-local” mechanism that drives the performance of the SDP relaxation. To the best of our knowledge, our result is the first exponentially decaying error bound for convex relaxations of mixture models. A corollary of our results shows that in certain regimes, the SDP solutions are in fact integral and exact. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of [Formula: see text] fraction of the points for any [Formula: see text]. As a special case, we show that under the [Formula: see text]-dimensional stochastic ball model, SDP achieves nontrivial (sometimes exact) recovery when the center separation is as small as [Formula: see text], which improves upon previous exact recovery results that require constant separation.

Private Weakly-Random Sequences from Human Heart Rate for Quantum Amplification.

We investigate whether the heart rate can be treated as a semi-random source with the aim of amplification by quantum devices. We use a semi-random source model called -Santha–Vazirani source, which can be amplified via quantum protocols to obtain a fully private random sequence. We analyze time intervals between consecutive heartbeats obtained from Holter electrocardiogram (ECG) recordings of people of different sex and age. We propose several transformations of the original time series into binary sequences. We have performed different statistical randomness tests and estimated quality parameters. We find that the heart can be treated as a good enough, and private by its nature, source of randomness that every human possesses. As such, in principle, it can be used as input to quantum device-independent randomness amplification protocols. The properly interpreted parameter can potentially serve as a new characteristic of the human heart from the perspective of medicine.

Sparse Subspace Clustering via Two-Step Reweighted L1-Minimization: Algorithm and Provable Neighbor Recovery Rates

Sparse subspace clustering (SSC) relies on sparse regression for accurate neighbor identification. Inspired by recent progress in compressive sensing, this paper proposes a new sparse regression scheme for SSC via two-step reweighted $\ell _{1} $ -minimization, which also generalizes a two-step $\ell _{1} $ -minimization algorithm introduced by E. J. Candes et al. in [ The Annals of Statistics , vol. 42, no. 2, pp. 669–699, 2014] without incurring extra algorithmic complexity. To fully exploit the prior information offered by the computed sparse representation vector in the first step, our approach places a weight on each component of the regression vector, and solves a weighted LASSO in the second step. We propose a data weighting rule suitable for enhancing neighbor identification accuracy. Then, under the formulation of the dual problem of weighted LASSO, we study in depth the theoretical neighbor recovery rates of the proposed scheme. Specifically, an interesting connection between the locations of nonzeros of the optimal sparse solution to the weighted LASSO and the indexes of the active constraints of the dual problem is established. Afterwards, under the semi-random model, analytic probability lower/upper bounds for various neighbor recovery events are derived. Our analytic results confirm that, with the aid of data weighting and if the prior neighbor information is accurate enough, the proposed scheme with a higher probability can produce many correct neighbors and few incorrect neighbors as compared to the solution without data weighting. Computer simulations are provided to validate our analytic study and evidence the effectiveness of the proposed approach.

Different models for simulation of mechanical behaviour of porous materials.

Commercially pure Titanium (cpTi) and its alloys are the most successful metallic biomaterials for bone replacement, due to its excellent biomechanical and biofunctional balance. However, these materials have higher elastic modulus when compared with bone, leading to the stress-shielding phenomenon and promoting bone resorption. Development of porous implants with low elastic modulus, providing a good mechanical and functional balance (suitable mechanical strength and optimum osseointegration), is the focus of emergent research in advanced Ti-based alloy biomaterials. With the aim of understanding the mechanical behaviour of porous materials with relation to the porosity level and the porous morphology, a new improved model with three different versions have been developed in this work. The proposed FE model combines the simplicity of a 2D periodic geometry with the complex information of the pore morphology extracted from experimentation. The methodology to generate the 2D simulated microstructure is based on a series of nxn pores distributed in a square matrix. The different versions of the model differ in the way of building the porous geometry. In the first version of the model ("Basic-Pattern Model"), the pores are supposed to be circular and periodically distributed in the matrix, following a perfect pattern. The second version of the model ("Pattern Model") is similar to the previous one, but with elliptic pores with a morphology randomly generated, following statistical information from experiments. In the third version ("Semi-random Model"), a controlled random distribution of the pores is obtained by including a randomness factors in both directions. By making use of the proposed FE model with its different versions, five different porous titanium obtained by the space-holders technique (with porosities θ = 28%, 37%, 47%, 57% and 66%) have been modeled based on experimental information of the pore morphology, and its macroscopic mechanical behaviour has been simulated, showing relatively good agreement with experimental results.

Endoscopic transsphenoidal surgery versus microsurgery for the resection of pituitary adenomas: a systematic review

To evaluate the efficacy and safety between endoscopic and microscopic surgery for transsphenoidal pituitary adenoma. Randomized or semi-random controlled trials comparing endoscopic with microscopic surgery for transsphenoidal pituitary adenoma and published between January 2000 and July 2013 were recruited. This meta-analysis (RevMan 5.1 software) was conducted to estimate short-term and long-term complications. Fixed random effect model or semi-random effect model was established to analyse the data. Twelve randomized or semi-random controlled trials were included in this study. Among 848 patients studied, 380 of them were treated with endoscopic and 468 were treated with microscopic. The analysis of the basic characteristics of these patients included in these studies showed that: compared with microscopic, the follow-up of patients in endoscopic group was shorter [OR = -2.29, 95%CI (-4.18, -0.39), P = 0.02], while there were no significant difference in gender proportion and the age of patients between the two groups (P > 0.05) . Also, compared with endoscopic, there was a significant addition in the incidence of diabetes insipidus [OR = 0.45, 95%CI(0.30,0.66), P < 0.0001] and other complications [OR = 0.26, 95%CI ( 0.12, 0.57), P = 0.0008] in the microscopic group, the blood loss during surgery was more [OR = -0.62, 95%CI(-1.19, 0.05), P = 0.03], the rate of complete tumor resection lower [OR = 0.61, 95%CI(0.39, 0.96), P = 0.03], and the hospitalization [OR = -1.53, 95%CI(-2.18, -0.88), P < 0.00001] was also significantly longer, there was no significant difference in the incidence of cerebrospinal fluid leakage, operation time and vision improvement rate (P > 0.05). For patients with pituitary tumors, endoscopic surgery may be more suitable compared with microscopic.

The smoothed complexity of edit distance

We initiate the study of the smoothed complexity of sequence alignment, by proposing a semi-random model of edit distance between two input strings, generated as follows: First, an adversary chooses two binary strings of length d and a longest common subsequence A of them. Then, every character is perturbed independently with probability p , except that A is perturbed in exactly the same way inside the two strings. We design two efficient algorithms that compute the edit distance on smoothed instances up to a constant factor approximation. The first algorithm runs in near-linear time, namely d {1+ϵ} for any fixed ϵ &gt; 0. The second one runs in time sublinear in d , assuming the edit distance is not too small. These approximation and runtime guarantees are significantly better than the bounds that were known for worst-case inputs. Our technical contribution is twofold. First, we rely on finding matches between substrings in the two strings, where two substrings are considered a match if their edit distance is relatively small, a prevailing technique in commonly used heuristics, such as PatternHunter of Ma et al. [2002]. Second, we effectively reduce the smoothed edit distance to a simpler variant of (worst-case) edit distance, namely, edit distance on permutations (a.k.a. Ulam's metric). We are thus able to build on algorithms developed for the Ulam metric, whose much better algorithmic guarantees usually do not carry over to general edit distance.

On the tractability of coloring semirandom graphs

As part of the efforts to understand the intricacies of the k-colorability problem, different distributions over k-colorable graphs have been analyzed. While the problem is notoriously hard (not even reasonably approximable) in the worst case, the average case (with respect to such distributions) often turns out to be “easy”. Semi-random models mediate between these two extremes and are more suitable to imitate “real-life” instances than purely random models. In this work we consider semi-random variants of the planted k-colorability distribution. This continues a line of research pursued by Coja-Oghlan, and by Krivelevich and Vilenchik. Our aim is to study a more general semi-random framework than those suggested so far. On the one hand we show that previous algorithmic techniques extend to our more general semi-random setting; on the other hand we give a hardness result, proving that a closely related semi-random model is intractable. Thus we provide some indication about which properties of the input distribution make the k-colorability problem hard.

Finding Large Independent Sets in Polynomial Expected Time

We consider instances of the maximum independent set problem that are constructed according to the following semirandom model. Let $G_{n,p}$ be a random graph, and let $S$ be a set of $k$ vertices, chosen uniformly at random. Then, let $G_0$ be the graph obtained by deleting all edges connecting two vertices in $S$. Finally, an adversary may add edges to $G_0$ that do not connect two vertices in $S$, thereby producing the instance $G=G_{n,p,k}^*$. We present an algorithm that on input $G=G_{n,p,k}^*$ finds an independent set of size $\geq k$ within polynomial expected time, provided that $k\geq C(n/p)^{1/2}$ for a certain constant $C>0$. Moreover, we prove that in the case $k\leq (1-\varepsilon)\ln(n)/p$ this problem is hard.

Heuristics for Semirandom Graph Problems

We consider semirandom graph models for finding large independent sets, colorings, and bisections in graphs. These models generate problem instances by blending random and adversarial decisions. To generate semirandom independent set problems, an independent set S of αn vertices is randomly chosen. Each edge connecting S with S is chosen with probability p, and an adversary is then allowed to add new edges arbitrarily, provided that S remains an independent set. The smaller p is, the greater the control the adversary has over the semirandom graph. We give a heuristic that with high probability recovers an independent set of size αn whenever p> (1+ε)lnn/αn, for any constant ε>0. We show that when p<(1−ε)lnn /αn, an independent set of size |S| cannot be recovered, unless NP⊆BPP. We use our result for maximum independent sets to obtain greatly improved heuristics for the model of k-colorable semirandom graphs introduced by Blum and Spencer. For constant k, our results are optimal up to constant factors in the edge probabilities. In the semirandom model for graph bisection, a random bisection (S, S) of the vertices is chosen. Each edge (u, v)∈S×S is independently chosen with probability q and each edge (u, v)∉S×S is independently chosen with probability p>q. The adversary may then arbitrarily remove edges in S×S and add edges not in S×S. Extending the work of Boppana, we give a heuristic that recovers this bisection with high probability when p−q⩾cplogn/n, for c a sufficiently large constant.

Minimum Coloring k-Colorable Graphs in Polynomial Average Time

We present algorithms for minimum (χ(G)-) coloring k-colorable graphs drawn from random and semi-random models. In both models, an adversary initially splits the vertices into k color classes V1,…,Vk of sizes Ω(n) each. In the first model, each potential edge joining vertices in different classes is included with independent probability p. In the latter model, the adversary fixes an ordering of all potential edges and then for each i, the ith edge (according to this ordering) is included with probability p(ei,S)≥p,S⊆{1,…,i−1}. Thus, the probability of inclusion for the ith edge depends on the outcomes on previous i−1 edges. Semi-random models were introduced as a way of striking a balance between random graphs and worst-case adversaries. Our algorithms run in polynomial time on average. χ(G)-coloring is harder than k-coloring because even a “short certificate” is not presently known for the optimality of a coloring. Our algorithms work as long as p≥n−α(k)+ϵ for the semi-random model and p≥n−γ(k)+ϵ for the random model where α(k)=(2k)/((k−1)(k+2)), γ(k)=(2k)/(k2−k+2), and ϵ is any positive constant.

Algorithms for coloring semi-random graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require $n^\delta (\delta>0)$ colors even for bounded chromatic k-colorable for fixed k n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms w always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19, 204-234 1995] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work $p \ge n^{-{\alpha (k)}+\epsilon}}$ Where $\alpha(k) = \frac{(2k)}{((k-1)(k+2))}$ and \epsilon is a positive constant.

Statistics of large-scale sequence searching.

Database search programs such as FASTA, BLAST or a rigorous Smith-Waterman algorithm produce lists of database entries, which are assumed to be related to the query. The computation of statistical significance of similarity scores is well established for single pairs of sequences and using purely random models. However, the multi-trial context of a database search poses new problems. The credibility of a certain score obtained in a database search decreases with the amount of data that is compared. To improve p-value computation for database search experiments, statistical properties of the databases, such as the distribution of sequence length and effects induced by frequently repeated sequence patterns, need to be taken into account. We investigated the SWISS-PROT protein database Release 31.0 running extensive simulations of database searches. A discrepancy is observed between the theoretical predictions and the empirical distribution. To correct for this, we evaluate the statistical significance of scores in the context of a database search by a contrasting semi-random model. This model enhances purely random models by one additional parameter reflecting individual statistical properties of real databases. We call this parameter the effective size of the database. r.spang@dkfz-heidelberg.de;m.vingron@dkfz-hei del berg.de