In the context of mining for frequent patterns using the standard levelwise algorithm, the following question arises: given the current level and the current set of frequent patterns, what is the maximal number of candidate patterns that can be generated on the next level? We answer this question by providing a tight upper bound, derived from a combinatorial result from the sixties by Kruskal and Katona. Our result is useful to reduce the number of database scans.
The sampling algorithm proposed by Toivonen @cite_3 performs at most two scans through the database by picking a random sample from the database, then finding all frequent patterns that probably hold in the whole database, and then verifying the results with the rest of the database. In the cases where the sampling method does not produce all frequent patterns, the missing patterns can be found by generating all remaining potentially frequent patterns and verifying their frequencies during a second pass through the database. The probability of such a failure can be kept small by decreasing the minimal support threshold. However, for a reasonably small probability of failure, the threshold must be drastically decreased, which can again cause a combinatorial explosion of the number of candidate patterns.
@cite_3
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A constructive solution of the irregular sampling problem for band- limited functions is given. We show how a band-limited function can be com- pletely reconstructed from any random sampling set whose density is higher than the Nyquist rate, and give precise estimates for the speed of convergence of this iteration method. Variations of this algorithm allow for irregular sampling with derivatives, reconstruction of band-limited functions from local averages, and irregular sampling of multivariate band-limited functions. In the irregular sampling problem one is asked whether and how a band- limited function can be completely reconstructed from its irregularly sam- pled values f(xi). This has many applications in signal and image processing, seismology, meteorology, medical imaging, etc. Finding constructive solutions of this problem has received considerable attention among mathematicians and engineers. The mathematical literature provides several uniqueness results (1, 2, 17, 18, 19). It is now part of the folklore that for stable sampling the sampling rate must be at least the Nyquist rate (18). These results, as deep as they are, have had little impact for the applied sciences, because they were not constructive. If the sampling set is just a perturbation of the regular oversampling, then a reconstruction method has been obtained in a seminal paper by Duffin and Schaeffer (6) (see also (29)): if for some L > 0, a > 0, and o > 0 the sampling points xk , k e Z , satisfy (a) - ok a, k ^ I, then the norm equivalence A iR (x) ) with w < n o. This norm equivalence implies that it is possible to reconstruct through an iterative procedure, the so-called frame method. Most of the later work on constructive methods consists of variations of this method (3, 21, 22, 26). The above conditions on the sampling set exclude random irregular sampling sets, e.g., sets with regions of higher sampling density. A partial, but undesirable remedy, to handle highly irregular sampling sets, would be to force the above conditions by throwing away information on part of the points and accept a very slow convergence of the iteration. We study the problem of sampling a random signal with sparse support in frequency domain. Shannon famously considered a scheme that instantaneously samples the signal at equispaced times. He proved that the signal can be reconstructed as long as the sampling rate exceeds twice the bandwidth (Nyquist rate). Candes, Romberg, Tao introduced a scheme that acquires instantaneous samples of the signal at random times. They proved that the signal can be uniquely and efficiently reconstructed, provided the sampling rate exceeds the frequency support of the signal, times logarithmic factors. In this paper we consider a probabilistic model for the signal, and a sampling scheme inspired by the idea of spatial coupling in coding theory. Namely, we propose to acquire non-instantaneous samples at random times. Mathematically, this is implemented by acquiring a small random subset of Gabor coefficients. We show empirically that this scheme achieves correct reconstruction as soon as the sampling rate exceeds the frequency support of the signal, thus reaching the information theoretic limit. In this paper, we study the problem of sampling from a given probability density function that is known to be smooth and strongly log-concave. We analyze several methods of approximate sampling based on discretizations of the (highly overdamped) Langevin diffusion and establish guarantees on its error measured in the Wasserstein-2 distance. Our guarantees improve or extend the state-of-the-art results in three directions. First, we provide an upper bound on the error of the first-order Langevin Monte Carlo (LMC) algorithm with optimized varying step-size. This result has the advantage of being horizon free (we do not need to know in advance the target precision) and to improve by a logarithmic factor the corresponding result for the constant step-size. Second, we study the case where accurate evaluations of the gradient of the log-density are unavailable, but one can have access to approximations of the aforementioned gradient. In such a situation, we consider both deterministic and stochastic approximations of the gradient and provide an upper bound on the sampling error of the first-order LMC that quantifies the impact of the gradient evaluation inaccuracies. Third, we establish upper bounds for two versions of the second-order LMC, which leverage the Hessian of the log-density. We nonasymptotic guarantees on the sampling error of these second-order LMCs. These guarantees reveal that the second-order LMC algorithms improve on the first-order LMC in ill-conditioned settings. 1. Introduction F. Marvasti. 2. An Introduction to Sampling Analysis P.L. Butzer, et al 3. Lagrange Interpolation and Sampling Theorems A.I. Zayed, P.L. Butzer. 4. Random Topics in Nonuniform Sampling F. Marvasti. 5. Iterative and Noniterative Recovery of Missing Samples for 1- D Band-Limited Signals P.J.S.G. Ferreira. 6. Numerical and Theoretical Aspects of Nonuniform Sampling of Band-Limited Images K. Grochenig, T. Strohmer. 7. The Nonuniform Discrete Fourier Transform S. Bagchi, S.K. Mitra. 8. Reconstruction of Stationary Processes Sampled at Random Times B. Lacaze. 9. Zero Crossings of Random Processes with Application to Estimation and Detection J. Barnett. 10. Magnetic Resonance Image Reconstruction from Nonuniformly Sampled k-Space Data F.T.A.W. Wajer, et al 11. Irregular and Sparse Sampling in Exploration Seismology A.J.W. Duijndam, et al 12. Randomized Digital Optimal Control W.L. de Koning, L.G. van Willigenburg. 13. Prediction of Band-Limited Signals from Past Samples and Applications to Speech Coding D.H. Muler, Y. Wu. 14. Frames, Irregular Sampling, and a Wavelet Auditory Model J.J. Benedetto, S. Scott. 15. Application of the Nonuniform Sampling to Motion Compensated Prediction A. Sharif, et al 16. Applications of Nonuniform Sampling to Nonlinear Modulation, A D and D A Techniques F. Marvasti, M. Sandler. 17. Applications to Error Correction Codes F. Marvasti. 18. Application of Nonuniform Sampling to Error Concealment M. Hasan, F. Marvasti. 19. Sparse Sampling in Array Processing S. Holm, et al 20. Fractional Delay Filters: Design and Applications V. Valimaki, T.I. Laakso.