7,160 research outputs found

    Learning Coverage Functions and Private Release of Marginals

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    We study the problem of approximating and learning coverage functions. A function c:2[n]R+c: 2^{[n]} \rightarrow \mathbf{R}^{+} is a coverage function, if there exists a universe UU with non-negative weights w(u)w(u) for each uUu \in U and subsets A1,A2,,AnA_1, A_2, \ldots, A_n of UU such that c(S)=uiSAiw(u)c(S) = \sum_{u \in \cup_{i \in S} A_i} w(u). Alternatively, coverage functions can be described as non-negative linear combinations of monotone disjunctions. They are a natural subclass of submodular functions and arise in a number of applications. We give an algorithm that for any γ,δ>0\gamma,\delta>0, given random and uniform examples of an unknown coverage function cc, finds a function hh that approximates cc within factor 1+γ1+\gamma on all but δ\delta-fraction of the points in time poly(n,1/γ,1/δ)poly(n,1/\gamma,1/\delta). This is the first fully-polynomial algorithm for learning an interesting class of functions in the demanding PMAC model of Balcan and Harvey (2011). Our algorithms are based on several new structural properties of coverage functions. Using the results in (Feldman and Kothari, 2014), we also show that coverage functions are learnable agnostically with excess 1\ell_1-error ϵ\epsilon over all product and symmetric distributions in time nlog(1/ϵ)n^{\log(1/\epsilon)}. In contrast, we show that, without assumptions on the distribution, learning coverage functions is at least as hard as learning polynomial-size disjoint DNF formulas, a class of functions for which the best known algorithm runs in time 2O~(n1/3)2^{\tilde{O}(n^{1/3})} (Klivans and Servedio, 2004). As an application of our learning results, we give simple differentially-private algorithms for releasing monotone conjunction counting queries with low average error. In particular, for any knk \leq n, we obtain private release of kk-way marginals with average error αˉ\bar{\alpha} in time nO(log(1/αˉ))n^{O(\log(1/\bar{\alpha}))}

    Reynolds Pressure and Relaxation in a Sheared Granular System

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    We describe experiments that probe the evolution of shear jammed states, occurring for packing fractions ϕSϕϕJ\phi_S \leq \phi \leq \phi_J, for frictional granular disks, where above ϕJ\phi_J there are no stress-free static states. We use a novel shear apparatus that avoids the formation of inhomogeneities known as shear bands. This fixed ϕ\phi system exhibits coupling between the shear strain, γ\gamma, and the pressure, PP, which we characterize by the `Reynolds pressure', and a `Reynolds coefficient', R(ϕ)=(2P/γ2)/2R(\phi) = (\partial ^2 P/\partial \gamma ^2)/2. RR depends only on ϕ\phi, and diverges as R(ϕcϕ)αR \sim (\phi_c - \phi)^{\alpha}, where ϕcϕJ\phi_c \simeq \phi_J, and α3.3\alpha \simeq -3.3. Under cyclic shear, this system evolves logarithmically slowly towards limit cycle dynamics, which we characterize in terms of pressure relaxation at cycle nn: ΔPβln(n/n0)\Delta P \simeq -\beta \ln(n/n_0). β\beta depends only on the shear cycle amplitude, suggesting an activated process where β\beta plays a temperature-like role.Comment: 4 pages, 4 figure

    On the design and implementation of broadcast and global combine operations using the postal model

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    There are a number of models that were proposed in recent years for message passing parallel systems. Examples are the postal model and its generalization the LogP model. In the postal model a parameter λ is used to model the communication latency of the message-passing system. Each node during each round can send a fixed-size message and, simultaneously, receive a message of the same size. Furthermore, a message sent out during round r will incur a latency of hand will arrive at the receiving node at round r + λ - 1. Our goal in this paper is to bridge the gap between the theoretical modeling and the practical implementation. In particular, we investigate a number of practical issues related to the design and implementation of two collective communication operations, namely, the broadcast operation and the global combine operation. Those practical issues include, for example, 1) techniques for measurement of the value of λ on a given machine, 2) creating efficient broadcast algorithms that get the latency hand the number of nodes n as parameters and 3) creating efficient global combine algorithms for parallel machines with λ which is not an integer. We propose solutions that address those practical issues and present results of an experimental study of the new algorithms on the Intel Delta machine. Our main conclusion is that the postal model can help in performance prediction and tuning, for example, a properly tuned broadcast improves the known implementation by more than 20%

    EEG Signal Analysis for Effective Classification of Brain States

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    EEG (Electroencephalogram) is a non-stationary signal that has been well established to be used for studying various states of the brain, in general, and several disorders, in particular. This work presents efficient signal processing and classification of the EEG signal. The digital filters used during decomposition of the input EEG signal have transfer functions which are simple and easily realizable on digital signal processors (DSP) and embedded systems. The features selected in this study; energy, entropy and variance; are among the most efficient and informative to analyze the EEG signal strength and distribution for detecting brain disorders such as seizure. Training and testing of the extracted features are performed using linear kernel (Support Vector Machine) SVM and thresholding in DSP algorithms and hardware, respectively. The experimental results for the digital signal processing algorithms show a high classification accuracy of 95% in the occurrence of seizure in epileptic patients. The techniques in this work are also under investigation for classifying other brain states/disorders such as sleep stages, sleep apnea and multiple sclerosis
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