deterministic wavelet thresholding for maximum-error metrics minos garofalakis bell laboratories...

Deterministic Wavelet Thresholding for Maximum-Error Metrics

Minos GarofalakisBell Laboratories Lucent Technologies 600 Mountain Avenue Murray Hill, NJ 07974Amit KumarDept. of Computer Science and Engineering Indian Institute of Technology, Delhi Hauz Khas, New Delhi 110 016, India

PODS 2004

Outline

Introduction Wavelet basics Deterministic wavelet thresholding

One-dimensional wavelet thresholding Multi-dimensional wavelet thresholding

Conclusions and future work

Introduction

Due to the exploratory nature of many today’s Decision Support Systems applications, there are a number of scenarios in which an exact answer may not be required, and a user may in fact prefer a fast, approximate answer. For example, during a drill-down query sequence in ad-hoc data mining.

Introduction Approximate answers obtained from

appropriate synopses of the data may be the only available option when the base data is remote and unavailable.

It is often the case that the full precision of the exact answer is not needed and the first few digits of precision will suffice.

Introduction The wavelet decomposition has

demonstrated the effectiveness in reducing large amounts of data to compact sets of wavelet coefficients (termed “wavelet synopses”) that can be used to provide fast and reasonably accurate approximate answers to queries.

Introduction A major shortcoming of conventional wavel

et-based techniques is the fact that the quality of approximate answers can vary widely and no meaningful error guarantees can be provided to the users.

we propose novel, computationally efficient schemes for deterministic wavelet thresholding with the objective of optimizing maximum-error metrics.

Wavelet basics

One-dimensional Haar Wavelets

Wavelet basics

Multi-dimensional Haar Wavelets

A=

Wavelet basics

Error tree for two dimension

Multi-dimension

Wavelet basics Coefficient Thresholding:

The goal of coecient thresholding is to determine the “best” subset of B coefficients to retain, so that some overall error measure in the approximation is minimized.

Conventional coefficient thresholding greedily retains the B largest Haar-wavelet coefficients in absolute normalized value.

Probabilistic thresholding schemes based on randomized rounding.

Deterministic wavelet thresholding Rather than trying to probabilistically contr

ol maximum relative error, a more direct solution would be to design deterministic thresholding schemes that explicitly minimize maximum-error metrics.

Such schemes can not only guarantee better synopses, but they can also avoid the potential pitfalls of randomized techniques, as well as the space-quantization requirement.

Deterministic wavelet thresholdingOne-dimensional wavelet thresholding

We propose a scheme based on Dynamic-Programming for building deterministic Haar-wavelet synopses that minimize the maximum error, which is defined as follows:

Deterministic wavelet thresholdingOne-dimensional wavelet thresholding

The basic idea is to condition the optimal error value for an error subtree not only on the root node cj of the subtree and the amount B of storage allotted, but on the error that “enters” that subtree through the coefficient selections made on the path from the root to node cj (excluding cj itself).

Let M[j; b; S] denote the minimum value of the maximum error (relative or absolute) among all data values in Tj .

Deterministic wavelet thresholdingOne-dimensional wavelet thresholding

M[j; b; S]

Path(c6)S Path(c6)

4

Deterministic wavelet thresholdingOne-dimensional wavelet thresholding

Deterministic wavelet thresholdingOne-dimensional wavelet thresholding

The base case (i.e. leaf)

In an internal node

Deterministic wavelet thresholdingOne-dimensional wavelet thresholding

Tine and space complexity: ,

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

It implies that the total number of possible ancestor subsets S for multi-dimensional coefficient at a level l is , rendering the exhaustive-enumeration DP scheme completely impractical.

We introduce efficient, polynomial-time approximation schemes for deterministic multi-dimensional wavelet thresholding for maximum-error metrics.

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding The key idea in our approximation scheme i

s to try to approximately “cover” the range of all possible error contributions for paths up to the root node of an error subtree using a much smaller number of (approximate) error values.

Let R denote the maximum absolute coefficient value in the error tree. The additive contribution to the absolute data-value reconstruction error from any possible path in the error tree is guaranteed to lie in the range

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

Our approximation scheme covers the entire R range using error-value breakpoints of the form , for a range of integer values for the exponent k.

Note that the number of such breakpoints needed is essentially

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

Let be a function that rounds any value down to the closest value in the set

Ma[j; b; e] capturing the approximate maximum error in the Tj error subtree , assuming a space budget of b coefficients allotted to Tj and an approximate/rounded additive error contribution of due to proper ancestors of node j being discarded from the synopsis.

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

The base case (i.e. leaf)

In an internal node

Where denote the sign of c’s contribution to the ji child subtree

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

To avoid the factor in run-time complexity. To generalize our approximate DP-array entries to

This generalization comes increase of in terms of time and space complexity.

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

The special case of minimizing absolute error is that all wavelet coefficient are integers.

Let Rz denote the maximum coefficient value in the error tree. The absolute error is guaranteed to line in the integer range

The key idea of the scheme for absolute error is then to intelligently scale-down the coefficients so that the possible range of integer additive-error values entering a subtree is polynomially- bounded.

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

For an internal node j, we compute as

Given a threshold parameter Let denote the set of coefficients with

absolute value greater than , and define as the quantity

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

The algorithm replaces each coefficient c in the error tree with a scaled-down coefficient value ,and work with these scaled coefficient.

A coefficient c is dropped if its scaled version satisfies

The range of possible incoming error is guaranteed to be only

Deterministic wavelet thresholdingMulti-dimensional wavelet thresholding

For a fixed parameter , the run time is only

Our absolute-error approximation scheme employs the algorithm for each value

Conclusions and future work We have proposed novel, computationally coeffici

ent schemes for deterministic maximum-error wavelet thresholding in one and multiple dimensions.

We are currently implementing our techniques and hope to report our experimental findings in the near future.

There are several interesting directions for future research in this area. As demonstrated in this paper, deterministic Haar-wavelet thresholding for maximum-error metrics becomes significantly more difficult as the data dimensionality increases