Above are papers related to the following paper: [Paper_ID] Paper #613409. 	[Title] Efficient Non-parametric Estimation of Probability Density Functions [Abstract] Accurate and fast estimation of probability density functions is crucial for satisfactory computational performance in many scientific problems. When the type of density is known a priori, then the problem becomes statistical estimation of parameters from the observed values. In the non-parametric case, usual estimators make use of kernel functions. If X j ; j = 1; 2; : : : ; n is a sequence of i.i.d. random variables with estimated probability density function f n , in the kernel method the computation of the values f n (X 1 ); f n (X 2 ); : : : ; f n (X n ) requires O(n 2 ) operations, since each kernel needs to be evaluated at every X j . We propose a sequence of special weight functions for the non-parametric estimation of f which requires almost linear time: if m is a slowly growing function that increases without bound with n, our method requires only O(m 2 n) arithmetic operations. We derive conditions for convergence under a number of metrics, which turn out to be similar to those required for the convergence of kernel based methods. We also discuss experiments on different distributions and compare the efficiency and the accuracy of our computations with kernel based estimators for various values of n and m. Please summarise all the given information.

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Above are papers related to the following paper: [Paper_ID] Paper #294030. 	[Title] MDL Learning of Probabilistic Neural Networks for Discrete Problem Domains [Abstract] Given a problem, a case-based reasoning (CBR) system will search its case memory and use the stored cases to find the solution, possibly modifying retrieved cases to adapt to the required input specifications. In discrete domains CBR reasoning can be based on a rigorous Bayesian probability propagation algorithm. Such a Bayesian CBR system can be implemented as a probabilistic feedforward neural network with one of the layers representing the cases. In this paper we introduce a Minimum Description Length (MDL) based learning algorithm to obtain the proper network structure with the associated conditional probabilities. This algorithm together with the resulting neural network implementation provide a massively parallel architecture for solving the efficiency bottleneck in case-based reasoning. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #910. 	[Title] Learning networks for face analysis and synthesis [Abstract] This paper presents an overview of the face-related projects in our group. The unifying theme underlying our work is the use of example-based learning methods for both analyzing and synthesizing face images. We label the example face images (and for the problem of face detection, "near miss" faces as well) with descriptive parameters for pose, expression, identity, and face vs. non-face. Then, by using example-based learning techniques, we develop networks for performing analysis tasks such as pose and expression estimation, face recognition, and face detection in cluttered scenes. In addition to these analysis applications, we show how the example-based technique can also be used as a novel method for image synthesis that is for computer graphics. Please summarise all the given information.

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Above are papers related to the following paper: [Paper_ID] Paper #289885. 	[Title] Learning in neural networks with Bayesian prototypes [Abstract] Given a set of samples of a probability distribution on a set of discrete random variables, we study the problem of constructing a good approximative neural network model of the underlying probability distribution. Our approach is based on an unsupervised learning scheme where the samples are first divided into separate clusters, and each cluster is then coded as a single vector. These Bayesian prototype vectors consist of conditional probabilities representing the attribute-value distribution inside the corresponding cluster. Using these prototype vectors, it is possible to model the underlying joint probability distribution as a simple Bayesian network (a tree), which can be realized as a feedforward neural network capable of probabilistic reasoning. In this framework, learning means choosing the size of the prototype set, partitioning the samples into the corresponding clusters, and constructing the cluster prototypes. We describe how the prototypes can be determined, given a partition of the samples, and present a method for evaluating the likelihood of the corresponding Bayesian tree. We also present a greedy heuristic for searching through the space of different partition schemes with different numbers of clusters, aiming at an optimal approximation of the probability distribution. Please summarise all the given information.

Above are papers related to the following paper: [Paper_ID] Paper #77758. 	[Title] Parzen. On estimation of a probability density function and mode. Annual Mathematical Statistics, 33:1065-1076, 1962. [Abstract] To apply the algorithm for classification we assign each class a separate set of codebook Gaussians. Each set is only trained with patterns from a single class. After having trained the codebook Gaussians, each set provides an estimate of the probability function of one class; just as with Parzen window estimation, we take as the estimate of the pattern distribution the average of all Gaussians in the set. Classification of a pattern may now be done by calculating the probability of each class at the respective sample point, and assigning to the pattern the class with the highest probability. Hence the whole codebook plays a role in the classification of patterns. This is not the case with regular classification schemes using codebooks. We have tested the classification scheme on several classification tasks including the two spiral problem. We compared our algorithm to various other classification algorithms and it came out second; the best algorithm for the applications is the Parzen window estimation. However, the computing time and memory for Parzen window estimation are excessive when compared to our algorithm, and hence, in practical situations, our algorithm is to be preferred. We have developed a fast algorithm which combines attractive properties of both Parzen window estimation and vector quantization. The scale parameter is tuned adaptively and, therefore, is not set in an ad hoc manner. It allows a classification strategy in which all the codebook vectors are taken into account. This yields better results than the standard vector quantization techniques. An interesting topic for further research is to use radially non-symmetric Gaussians. Please summarise all the given information.

Paper #613409 has the title `Efficient Non-parametric Estimation of Probability Density Functions  
` and the abstract `Accurate and fast estimation of probability density functions is crucial for satisfactory computational performance in many scientific problems. When the type of density is known a priori, then the problem becomes statistical estimation of parameters from the observed values. In the non-parametric case, usual estimators make use of kernel functions. If X j ; j = 1; 2; : : : ; n is a sequence of i.i.d. random variables with estimated probability density function f n , in the kernel method the computation of the values f n (X 1 ); f n (X 2 ); : : : ; f n (X n ) requires O(n 2 ) operations, since each kernel needs to be evaluated at every X j . We propose a sequence of special weight functions for the non-parametric estimation of f which requires almost linear time: if m is a slowly growing function that increases without bound with n, our method requires only O(m 2 n) arithmetic operations. We derive conditions for convergence under a number of metrics, which turn out to be similar to those required for the convergence of kernel based methods. We also discuss experiments on different distributions and compare the efficiency and the accuracy of our computations with kernel based estimators for various values of n and m. 
`.
Question: Which category should Paper #613409 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].


Paper #613409 has the title `Efficient Non-parametric Estimation of Probability Density Functions  
` and the abstract `Accurate and fast estimation of probability density functions is crucial for satisfactory computational performance in many scientific problems. When the type of density is known a priori, then the problem becomes statistical estimation of parameters from the observed values. In the non-parametric case, usual estimators make use of kernel functions. If X j ; j = 1; 2; : : : ; n is a sequence of i.i.d. random variables with estimated probability density function f n , in the kernel method the computation of the values f n (X 1 ); f n (X 2 ); : : : ; f n (X n ) requires O(n 2 ) operations, since each kernel needs to be evaluated at every X j . We propose a sequence of special weight functions for the non-parametric estimation of f which requires almost linear time: if m is a slowly growing function that increases without bound with n, our method requires only O(m 2 n) arithmetic operations. We derive conditions for convergence under a number of metrics, which turn out to be similar to those required for the convergence of kernel based methods. We also discuss experiments on different distributions and compare the efficiency and the accuracy of our computations with kernel based estimators for various values of n and m. 
`.
Question: Which category should Paper #613409 be classified as ?
You can select one from ['Neural_Networks', 'Case_Based', 'Theory', 'Reinforcement_Learning', 'Probabilistic_Methods', 'Rule_Learning', 'Genetic_Algorithms'].

This paper should be classified into: Probabilistic_Methods