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In mathematics, a combination is a selection of items from a set that has distinct members, such that the order of selection does not matter (unlike permutations). For example, given three fruits, say an apple, an orange and a pear, there are three combinations of two that can be drawn from this set: an apple and a pear; an apple and an orange; or a pear and an orange. More formally, a k-combination of a set S is a subset of k distinct elements of S. So, two combinations are identical if and only if each combination has the same members. (The arrangement of the members in each set does not matter.) If the set has n elements, the number of k-combinations, denoted by C(n,k) or C^n_k, is equal to the binomial coefficient \binom nk = \frac{n(n-1)\dotsb(n-k+1)}{k(k-1)\dotsb1}, which can be written using factorials as \textstyle\frac{n!}{k!(n-k)!} whenever k\leq n, and which is zero when k>n. Th

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Multi-level preconditioner for solving the Navier-Stokes equations in hemodynamics applications

Dana Christen

We present a multi-level algorithm to approximate the inverse of the fluid block in a Navier-Stokes saddle-point matrix where the coarse level is defined as a restriction of the degrees of freedom to those of lower order finite elements. A one-level scheme involving P1 and P2 finite elements is studied in details and several transfer operators are compared by means of two reference problems. Numerical results show that restriction and prolongation operators based on L2 projection lead to faster GMRES convergence of the fluid part, for all the examined combinations of mesh size, time step and Reynolds number.

2013

Robust speech recognition based on multi-stream processing

Astrid Hagen

Despite sophisticated present day automatic speech recognition (ASR) techniques, a single recognizer is usually incapable of accounting for the varying conditions in a typical natural environment. Higher robustness to a range of noise cases can potentially be achieved by combining the results of several recognizers operating in parallel. One such approach is multi-band processing, mimicking parallel processing of frequency subbands in human speech recognition as had been claimed by Fletcher. However, recent findings in both human and automatic speech recognition have revealed insufficiencies, such as the assumption of independence between frequency subbands, of the original multi-band ASR approach which often leads to reduced performance in the case of clean speech and wide-band noise. To overcome this problem, we propose and investigate a new set of "full combination" rules which integrate acoustic models trained on all possible combinations of subbands, preserving correlation information and leading to higher performance in all noise conditions. In this development, particular attention was given to the theoretical basis for all of the rules developed in terms of statistical theory, so that the assumptions that were necessary in each model become clear. The new combination strategies are developed for both posterior- and likelihood-based systems. These new combination strategies are then also applied to the combination of diverse feature streams, for example derived from multi-time scale analysis, which results in better exploitation of the often used instantaneous and time difference features. While combination may give the same weight to each expert, robustness of a multiple stream system can be further enhanced when each stream expert is assigned a weight reflecting its reliability. The new combination techniques are tested with several fixed and adaptive weighting strategies, including relative frequency of correct classification, least mean squared error, local signal-to-noise ratio, and maximum-likelihood based weights. We will see how the new multi-band approaches, which are consistently trained in clean speech, outperform original multi-band ASR models in both clean and noisy speech. Multi-band processing improves over the baseline fullband recognizer only in the case of narrow-band noise. However, combining multiple data streams from different time scales, using the same "full combination" rules, has also shown to significantly improve over the baseline in wide-band factory noise.

EPFL2002

Robust speech recognition based on multi-stream processing

Astrid Hagen

full combination'' rules which integrate acoustic models trained on all possible combinations of subbands, preserving correlation information and leading to higher performance in all noise conditions. In this development, particular attention was given to the theoretical basis for all of the rules developed in terms of statistical theory, so that the assumptions that were necessary in each model become clear. The new combination strategies are developed for both posterior- and likelihood-based systems. These new combination strategies are then also applied to the combination of diverse feature streams, for example derived from multi-time scale analysis, which results in better exploitation of the often used instantaneous and time difference features. While combination may give the same weight to each expert, robustness of a multiple stream system can be further enhanced when each stream expert is assigned a weight reflecting its reliability. The new combination techniques are tested with several fixed and adaptive weighting strategies, including relative frequency of correct classification, least mean squared error, local signal-to-noise ratio, and maximum-likelihood based weights. We will see how the new multi-band approaches, which are consistently trained in clean speech, outperform original multi-band ASR models in both clean and noisy speech. Multi-band processing improves over the baseline fullband recognizer only in the case of narrow-band noise. However, combining multiple data streams from different time scales, using the same

full combination'' rules, has also shown to significantly improve over the baseline in wide-band factory noise.

École Polytechnique Fédérale de Lausanne2001

Binomial coefficient

In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 a

Permutation

In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The w

Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers le

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