High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length $n$, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity $\exp(\tilde{O}(\sqrt{\log n}))$. Previously such codes were known to exist only with $\Omega(n^{\beta})$ query complexity (for constant $\beta > 0$), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby~\cite{AL96_codes}. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant relative distance}. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity $\exp(\tilde{O}(\sqrt{\log n}))$, which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with $\exp(\tilde{O}(\sqrt{\log n}))$ query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any $o(n)$ query complexity. Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters

What does the local structure of a planar graph tell us about its global structure?

The local k-neighborhood of a vertex v in an unweighted graph G = (V,E) with vertex set V and edge set E is the subgraph induced by all vertices of distance at most k from v. The rooted k-neighborhood of v is also called a k-disk around vertex v. If a graph has maximum degree bounded by a constant d, and k is also constant, the number of isomorphism classes of k-disks is constant as well. We can describe the local structure of a bounded-degree graph G by counting the number of isomorphic copies in G of each possible k-disk. We can summarize this information in form of a vector that has an entry for each isomorphism class of k-disks. The value of the entry is the number of isomorphic copies of the corresponding k-disk in G. We call this vector frequency vector of k-disks. If we only know this vector, what does it tell us about the structure of G? In this paper we will survey a series of papers in the area of Property Testing that leads to the following result (stated informally): There is a k = k(ε,d) such that for any planar graph G its local structure (described by the frequency vector of k-disks) determines G up to insertion and deletion of at most εd n edges (and relabelling of vertices)

Bond-Propagation Algorithm for Thermodynamic Functions in General 2D Ising Models

Recently, we developed and implemented the bond propagation algorithm for calculating the partition function and correlation functions of random bond Ising models in two dimensions. The algorithm is the fastest available for calculating these quantities near the percolation threshold. In this paper, we show how to extend the bond propagation algorithm to directly calculate thermodynamic functions by applying the algorithm to derivatives of the partition function, and we derive explicit expressions for this transformation. We also discuss variations of the original bond propagation procedure within the larger context of Y-Delta-Y-reducibility and discuss the relation of this class of algorithm to other algorithms developed for Ising systems. We conclude with a discussion on the outlook for applying similar algorithms to other models.Comment: 12 pages, 10 figures; submitte

Nuclear-spin relaxation of 207^{207}Pb in ferroelectric powders

Motivated by a recent proposal by O. P. Sushkov and co-workers to search for a P,T-violating Schiff moment of the $^{207}$Pb nucleus in a ferroelectric solid, we have carried out a high-field nuclear magnetic resonance study of the longitudinal and transverse spin relaxation of the lead nuclei from room temperature down to 10 K for powder samples of lead titanate (PT), lead zirconium titanate (PZT), and a PT monocrystal. For all powder samples and independently of temperature, transverse relaxation times were found to be $T_2\approx 1.5$ms, while the longitudinal relaxation times exhibited a temperature dependence, with $T_1$ of over an hour at the lowest temperatures, decreasing to $T_1\approx 7$s at room temperature. At high temperatures, the observed behavior is consistent with a two-phonon Raman process, while in the low temperature limit, the relaxation appears to be dominated by a single-phonon (direct) process involving magnetic impurities. This is the first study of temperature-dependent nuclear-spin relaxation in PT and PZT ferroelectrics at such low temperatures. We discuss the implications of the results for the Schiff-moment search.Comment: 6 pages, 4 figure

Exploring Interpretability for Predictive Process Analytics

Modern predictive analytics underpinned by machine learning techniques has become a key enabler to the automation of data-driven decision making. In the context of business process management, predictive analytics has been applied to making predictions about the future state of an ongoing business process instance, for example, when will the process instance complete and what will be the outcome upon completion. Machine learning models can be trained on event log data recording historical process execution to build the underlying predictive models. Multiple techniques have been proposed so far which encode the information available in an event log and construct input features required to train a predictive model. While accuracy has been a dominant criterion in the choice of various techniques, they are often applied as a black-box in building predictive models. In this paper, we derive explanations using interpretable machine learning techniques to compare and contrast the suitability of multiple predictive models of high accuracy. The explanations allow us to gain an understanding of the underlying reasons for a prediction and highlight scenarios where accuracy alone may not be sufficient in assessing the suitability of techniques used to encode event log data to features used by a predictive model. Findings from this study motivate the need and importance to incorporate interpretability in predictive process analytics.Comment: 15 pages, 7 figure

Maximum likelihood drift estimation for a threshold diffusion

We study the maximum likelihood estimator of the drift parameters of a stochastic differential equation, with both drift and diffusion coefficients constant on the positive and negative axis, yet discontinuous at zero. This threshold diffusion is called drifted Oscillating Brownian motion.For this continuously observed diffusion, the maximum likelihood estimator coincide with a quasi-likelihood estimator with constant diffusion term. We show that this estimator is the limit, as observations become dense in time, of the (quasi)-maximum likelihood estimator based on discrete observations. In long time, the asymptotic behaviors of the positive and negative occupation times rule the ones of the estimators. Differently from most known results in the literature, we do not restrict ourselves to the ergodic framework: indeed, depending on the signs of the drift, the process may be ergodic, transient or null recurrent. For each regime, we establish whether or not the estimators are consistent; if they are, we prove the convergence in long time of the properly rescaled difference of the estimators towards a normal or mixed normal distribution. These theoretical results are backed by numerical simulations

Small grid embeddings of 3-polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value $\phi(S_T)$ at expiry $T$ if neither of the continuous time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time interval $[0,T]$. Using a probabilistic approach we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions $\phi$, barrier functions $b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier premium can be expressed as a sum of integrals along the barriers $b_\pm$ of the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic

Favorable outcome of early treatment of new onset child and adolescent migraine-implications for disease modification.

There is evidence that the prevalence of migraine in children and adolescents may be increasing. Current theories of migraine pathophysiology in adults suggest activation of central cortical and brainstem pathways in conjunction with the peripheral trigeminovascular system, which ultimately results in release of neuropeptides, facilitation of central pain pathways, neurogenic inflammation surrounding peripheral vessels, and vasodilatation. Although several risk factors for frequent episodic, chronic, and refractory migraine have been identified, the causes of migraine progression are not known. Migraine pathophysiology has not been fully evaluated in children. In this review, we will first discuss the evidence that early therapeutic interventions in the child or adolescent new onset migraineur, may halt or limit progression and disability. We will then review the evidence suggesting that many adults with chronic or refractory migraine developed their migraine as children or adolescents and may not have been treated adequately with migraine-specific therapy. Finally, we will show that early, appropriate and optimal treatment of migraine during childhood and adolescence may result in disease modification and prevent progression of this disease