64 research outputs found
Integrals of monomials over the orthogonal group
A recursion formula is derived which allows to evaluate invariant integrals
over the orthogonal group O(N), where the integrand is an arbitrary finite
monomial in the matrix elements of the group. The value of such an integral is
expressible as a finite sum of partial fractions in . The recursion formula
largely extends presently available integration formulas for the orthogonal
group.Comment: 9 pages, no figure
A computer algebra user interface manifesto
Many computer algebra systems have more than 1000 built-in functions, making
expertise difficult. Using mock dialog boxes, this article describes a proposed
interactive general-purpose wizard for organizing optional transformations and
allowing easy fine grain control over the form of the result even by amateurs.
This wizard integrates ideas including:
* flexible subexpression selection;
* complete control over the ordering of variables and commutative operands,
with well-chosen defaults;
* interleaving the choice of successively less main variables with applicable
function choices to provide detailed control without incurring a combinatorial
number of applicable alternatives at any one level;
* quick applicability tests to reduce the listing of inapplicable
transformations;
* using an organizing principle to order the alternatives in a helpful
manner;
* labeling quickly-computed alternatives in dialog boxes with a preview of
their results,
* using ellipsis elisions if necessary or helpful;
* allowing the user to retreat from a sequence of choices to explore other
branches of the tree of alternatives or to return quickly to branches already
visited;
* allowing the user to accumulate more than one of the alternative forms;
* integrating direct manipulation into the wizard; and
* supporting not only the usual input-result pair mode, but also the useful
alternative derivational and in situ replacement modes in a unified window.Comment: 38 pages, 12 figures, to be published in Communications in Computer
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Appropriate, accessible and appealing probabilistic graphical models
Appropriate - Many multivariate probabilistic models either use independent distributions or dependent Gaussian distributions. Yet, many real-world datasets contain count-valued or non-negative skewed data, e.g. bag-of-words text data and biological sequencing data. Thus, we develop novel probabilistic graphical models for use on count-valued and non-negative data including Poisson graphical models and multinomial graphical models. We develop one generalization that allows for triple-wise or k-wise graphical models going beyond the normal pairwise formulation. Furthermore, we also explore Gaussian-copula graphical models and derive closed-form solutions for the conditional distributions and marginal distributions (both before and after conditioning). Finally, we derive mixture and admixture, or topic model, generalizations of these graphical models to introduce more power and interpretability.
Accessible - Previous multivariate models, especially related to text data, often have complex dependencies without a closed form and require complex inference algorithms that have limited theoretical justification. For example, hierarchical Bayesian models often require marginalizing over many latent variables. We show that our novel graphical models (even the k-wise interaction models) have simple and intuitive estimation procedures based on node-wise regressions that likely have similar theoretical guarantees as previous work in graphical models. For the copula-based graphical models, we show that simple approximations could still provide useful models; these copula models also come with closed-form conditional and marginal distributions, which make them amenable to exploratory inspection and manipulation. The parameters of these models are easy to interpret and thus may be accessible to a wide audience.
Appealing - High-level visualization and interpretation of graphical models with even 100 variables has often been difficult even for a graphical model expert---despite visualization being one of the original motivators for graphical models. This difficulty is likely due to the lack of collaboration between graphical model experts and visualization experts. To begin bridging this gap, we develop a novel "what if?" interaction that manipulates and leverages the probabilistic power of graphical models. Our approach defines: the probabilistic mechanism via conditional probability; the query language to map text input to a conditional probability query; and the formal underlying probabilistic model. We then propose to visualize these query-specific probabilistic graphical models by combining the intuitiveness of force-directed layouts with the beauty and readability of word clouds, which pack many words into valuable screen space while ensuring words do not overlap via pixel-level collision detection. Although both the force-directed layout and the pixel-level packing problems are challenging in their own right, we approximate both simultaneously via adaptive simulated annealing starting from careful initialization. For visualizing mixture distributions, we also design a meaningful mapping from the properties of the mixture distribution to a color in the perceptually uniform CIELUV color space. Finally, we demonstrate our approach via illustrative visualizations of several real-world datasets.Computer Science
Advanced Probabilistic Models for Clustering and Projection
Probabilistic modeling for data mining and machine learning problems is a fundamental research area. The general approach is to assume a generative model underlying the observed data, and estimate model parameters via likelihood maximization. It has the deep probability theory as the mathematical background, and enjoys a large amount of methods from statistical learning, sampling theory and Bayesian statistics. In this thesis we study several advanced probabilistic models for data clustering and feature projection, which are the two important unsupervised learning problems.
The goal of clustering is to group similar data points together to uncover the data clusters. While numerous methods exist for various clustering tasks, one important question still remains, i.e., how to automatically determine the number of clusters. The first part of the thesis answers this question from a mixture modeling perspective. A finite mixture model is first introduced for clustering, in which each mixture component is assumed to be an exponential family distribution for generality. The model is then extended to an infinite mixture model, and its strong connection to Dirichlet process (DP) is uncovered which is a non-parametric Bayesian framework. A variational Bayesian algorithm called VBDMA is derived from this new insight to learn the number of clusters automatically, and empirical studies on some 2D data sets and an image data set verify the effectiveness of this algorithm.
In feature projection, we are interested in dimensionality reduction and aim to find a low-dimensional feature representation for the data. We first review the well-known principal component analysis (PCA) and its probabilistic interpretation (PPCA), and then generalize PPCA to a novel probabilistic model which is able to handle non-linear projection known as kernel PCA. An expectation-maximization (EM) algorithm is derived for kernel PCA such that it is fast and applicable to large data sets. Then we propose a novel supervised projection method called MORP, which can take the output information into account in a supervised learning context. Empirical studies on various data sets show much better results compared to unsupervised projection and other supervised projection methods. At the end we generalize MORP probabilistically to propose SPPCA for supervised projection, and we can also naturally extend the model to S2PPCA which is a semi-supervised projection method. This allows us to incorporate both the label information and the unlabeled data into the projection process.
In the third part of the thesis, we introduce a unified probabilistic model which can handle data clustering and feature projection jointly. The model can be viewed as a clustering model with projected features, and a projection model with structured documents. A variational Bayesian learning algorithm can be derived, and it turns out to iterate the clustering operations and projection operations until convergence. Superior performance can be obtained for both clustering and projection
Quantum Statistical Field Theory and Combinatorics
This is a set of review notes on combinatorial aspects of Bosonic quantum
field theory. We collect together several related issues concerning moments of
distributions, moments of stochastic processes and Ito's formula, and Green's
functions and cumulant moments in quantum field theory.Comment: 50 pages, several figures, extended notes with up-dated reference
Formalized Class Group Computations and Integral Points on Mordell Elliptic Curves
Diophantine equations are a popular and active area of research in number
theory. In this paper we consider Mordell equations, which are of the form
, where is a (given) nonzero integer number and all solutions in
integers and have to be determined. One non-elementary approach for
this problem is the resolution via descent and class groups. Along these lines
we formalized in Lean 3 the resolution of Mordell equations for several
instances of . In order to achieve this, we needed to formalize several
other theories from number theory that are interesting on their own as well,
such as ideal norms, quadratic fields and rings, and explicit computations of
the class number. Moreover we introduced new computational tactics in order to
carry out efficiently computations in quadratic rings and beyond.Comment: 14 pages. Submitted to CPP '23. Source code available at
https://github.com/lean-forward/class-group-and-mordell-equatio
Physics-inspired Replica Approaches to Computer Science Problems
We study machine learning class classification problems and combinatorial optimization problems using physics inspired replica approaches. In the current work, we focus on the traveling salesman problem which is one of the most famous problems in the entire field of combinatorial optimization. Our approach is specifically motivated by the desire to avoid trapping in metastable local minima-a common occurrence in hard problems with multiple extrema. Our method involves (i) coupling otherwise independent simulations of a system (“replicas”) via geometrical distances as well as (ii) probabilistic inference applied to the solutions found by individual replicas. In particular, we apply our method to the well-known “k-opt” algorithm and examine two particular cases-k = 2 and k = 3. With the aid of geometrical coupling alone, we are able to determine for the optimum tour length on systems up to 280 cities (an order of magnitude larger than the largest systems typically solved by the bare k = 3 opt). The probabilistic replica-based inference approach improves k - opt even further and determines the optimal solution of a problem with 318 cities. In this work, we also formulate a supervised machine learning algorithm for classification problems which is called “Stochastic Replica Voting Machine” (SRVM). The method is based on the representations of known data via multiple linear expansions in terms of various stochastic functions. The algorithm is developed, implemented and applied to a binary and a 3-class classification problems in material science. Here, we employ SRVM to predict candidate compounds capable of forming cubic Perovskite structure and further classify binary (AB) solids. We demonstrated that our SRVM method exceeds the well-known Support Vector Machine (SVM) in terms of accuracy when predicting the cubic Perovskite structure. The algorithm has also been tested on 8 diverse training data sets of various types and feature space dimensions from UCI machine learning repository. It has been shown to consistently match or exceed the accuracy of existing algorithms, while simultaneously avoiding many of their pitfalls
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