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Information Theory For Data Management. Divesh Srivastava Suresh Venkatasubramanian. Motivation. Information Theory is relevant to all of humanity. -- Abstruse Goose (177). Information Theory for Data Management - Divesh & Suresh. Background.

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information theory for data management

Information Theory For Data Management

Divesh Srivastava

Suresh Venkatasubramanian

motivation
Motivation

Information Theory is relevant to all of humanity...

-- Abstruse Goose (177)

Information Theory for Data Management - Divesh & Suresh

background
Background
  • Many problems in data management need precise reasoning about information content, transfer and loss
    • Structure Extraction
    • Privacy preservation
    • Schema design
    • Probabilistic data ?

Information Theory for Data Management - Divesh & Suresh

information theory
Information Theory
  • First developed by Shannon as a way of quantifying capacity of signal channels.
  • Entropy, relative entropy and mutual information capture intrinsic informational aspects of a signal
  • Today:
    • Information theory provides a domain-independent way to reason about structure in data
    • More information = interesting structure
    • Less information linkage = decoupling of structures

Information Theory for Data Management - Divesh & Suresh

tutorial thesis
Tutorial Thesis

Information theory provides a mathematical framework for the quantification of information content, linkage and loss.

This framework can be used in the design of data management strategies that rely on probing the structure of information in data.

Information Theory for Data Management - Divesh & Suresh

tutorial goals
Tutorial Goals
  • Introduce information-theoretic concepts to DB audience
  • Give a ‘data-centric’ perspective on information theory
  • Connect these to applications in data management
  • Describe underlying computational primitives

Illuminate when and how information theory might be of use in new areas of data management.

Information Theory for Data Management - Divesh & Suresh

outline
Outline

Part 1

Introduction to Information Theory

Application: Data Anonymization

Application: Database Design

Part 2

Review of Information Theory Basics

Application: Data Integration

Computing Information Theoretic Primitives

Open Problems

Information Theory for Data Management - Divesh & Suresh

histograms and discrete distributions

X

f(X)

X

p(X)

X

x1

4

x1

0.5

aggregate counts

x1

x2

2

x2

0.25

normalize

x3

1

x3

0.125

x3

x4

1

x4

0.125

x2

x4

Probability distribution

Histogram

x1

x1

Column of data

x2

x1

Histograms And Discrete Distributions

Information Theory for Data Management - Divesh & Suresh

histograms and discrete distributions9

X

f(X)

X

p(X)

X

x1

4

x1

0.667

aggregate counts

x1

x2

2

x2

0.2

x3

1

x3

0.067

x3

x4

1

x4

0.067

x2

x4

Probability distribution

Histogram

x1

x1

Column of data

x2

x1

Histograms And Discrete Distributions

reweight

normalize

Information Theory for Data Management - Divesh & Suresh

from columns to random variables
From Columns To Random Variables
  • We can think of a column of data as “represented” by a random variable:
    • X is a random variable
    • p(X) is the column of probabilities p(X = x1), p(X = x2), and so on
    • Also known (in unweighted case) as the empirical distribution induced by the column X.
  • Notation:
    • X (upper case) denotes a random variable (column)
    • x (lower case) denotes a value taken by X (field in a tuple)
    • p(x) is the probability p(X = x)

Information Theory for Data Management - Divesh & Suresh

joint distributions
Joint Distributions

Discrete distribution: probability p(X,Y,Z)

p(Y) = ∑x p(X=x,Y) = ∑x ∑z p(X=x,Y,Z=z)

Information Theory for Data Management - Divesh & Suresh

entropy of a column
Let h(x) = log2 1/p(x)

h(X) is column of h(x) values.

H(X) = EX[h(x)] = SX p(x) log2 1/p(x)

Two views of entropy

It captures uncertainty in data: high entropy, more unpredictability

It captures information content: higher entropy, more information.

Entropy Of A Column

H(X) = 1.75 < log |X| = 2

Information Theory for Data Management - Divesh & Suresh

examples
Examples
  • X uniform over [1, ..., 4]. H(X) = 2
  • Y is 1 with probability 0.5, in [2,3,4] uniformly.
    • H(Y) = 0.5 log 2 + 0.5 log 6 ~= 1.8 < 2
    • Y is more sharply defined, and so has less uncertainty.
  • Z uniform over [1, ..., 8]. H(Z) = 3 > 2
    • Z spans a larger range, and captures more information

X

Y

Z

Information Theory for Data Management - Divesh & Suresh

comparing distributions
Comparing Distributions
  • How do we measure difference between two distributions ?
  • Kullback-Leibler divergence:
    • dKL(p, q) = Ep[ h(q) – h(p) ] = Si pi log(pi/qi)

Inference mechanism

Prior belief

Resulting belief

Information Theory for Data Management - Divesh & Suresh

comparing distributions15
Comparing Distributions
  • Kullback-Leibler divergence:
    • dKL(p, q) = Ep[ h(q) – h(p) ] = Si pi log(pi/qi)
    • dKL(p, q) >= 0
    • Captures extra information needed to capture p given q
    • Is asymmetric ! dKL(p, q) != dKL(q, p)
    • Is not a metric (does not satisfy triangle inequality)
  • There are other measures:
    • 2-distance, variational distance, f-divergences, …

Information Theory for Data Management - Divesh & Suresh

conditional probability
Conditional Probability
  • Given a joint distribution on random variables X, Y, how much information about X can we glean from Y ?
  • Conditional probability: p(X|Y)
    • p(X = x1 | Y = y1) = p(X = x1, Y = y1)/p(Y = y1)

Information Theory for Data Management - Divesh & Suresh

conditional entropy
Conditional Entropy
  • Let h(x|y) = log2 1/p(x|y)
  • H(X|Y) = Ex,y[h(x|y)] = SxSy p(x,y) log2 1/p(x|y)
  • H(X|Y) = H(X,Y) – H(Y)
  • H(X|Y) = H(X,Y) – H(Y) = 2.25 – 1.5 = 0.75
  • If X, Y are independent, H(X|Y) = H(X)

Information Theory for Data Management - Divesh & Suresh

mutual information
Mutual Information
  • Mutual information captures the difference between the joint distribution on X and Y, and the marginal distributions on X and Y.
  • Let i(x;y) = log p(x,y)/p(x)p(y)
  • I(X;Y) = Ex,y[I(X;Y)] = SxSy p(x,y) log p(x,y)/p(x)p(y)

Information Theory for Data Management - Divesh & Suresh

mutual information strength of linkage
Mutual Information: Strength of linkage
  • I(X;Y) = H(X) + H(Y) – H(X,Y) = H(X) – H(X|Y) = H(Y) – H(Y|X)
  • If X, Y are independent, then I(X;Y) = 0:
    • H(X,Y) = H(X) + H(Y), so I(X;Y) = H(X) + H(Y) – H(X,Y) = 0
  • I(X;Y) <= max (H(X), H(Y))
    • Suppose Y = f(X) (deterministically)
    • Then H(Y|X) = 0, and so I(X;Y) = H(Y) – H(Y|X) = H(Y)
  • Mutual information captures higher-order interactions:
    • Covariance captures “linear” interactions only
    • Two variables can be uncorrelated (covariance = 0) and have nonzero mutual information:
    • X R [-1,1], Y = X2. Cov(X,Y) = 0, I(X;Y) = H(X) > 0

Information Theory for Data Management - Divesh & Suresh

information theory summary
Information Theory: Summary
  • We can represent data as discrete distributions (normalized histograms)
  • Entropy captures uncertainty or information content in a distribution
  • The Kullback-Leibler distance captures the difference between distributions
  • Mutual information and conditional entropy capture linkage between variables in a joint distribution

Information Theory for Data Management - Divesh & Suresh

outline21
Outline

Part 1

Introduction to Information Theory

Application: Data Anonymization

Application: Database Design

Part 2

Review of Information Theory Basics

Application: Data Integration

Computing Information Theoretic Primitives

Open Problems

Information Theory for Data Management - Divesh & Suresh

data anonymization using randomization
Data Anonymization Using Randomization

Goal: publish anonymized microdata to enable accurate ad hoc analyses, but ensure privacy of individuals’ sensitive attributes

Key ideas:

Randomize numerical data: add noise from known distribution

Reconstruct original data distribution using published noisy data

Issues:

How can the original data distribution be reconstructed?

What kinds of randomization preserve privacy of individuals?

Information Theory for Data Management - Divesh & Suresh

data anonymization using randomization23
Data Anonymization Using Randomization

Many randomization strategies proposed [AS00, AA01, EGS03]

Example randomization strategies: X in [0, 10]

R = X + μ (mod 11), μ is uniform in {-1, 0, 1}

R = X + μ (mod 11), μ is in {-1 (p = 0.25), 0 (p = 0.5), 1 (p = 0.25)}

R = X (p = 0.6), R = μ, μ is uniform in [0, 10] (p = 0.4)

Question:

Which randomization strategy has higher privacy preservation?

Quantify loss of privacy due to publication of randomized data

Information Theory for Data Management - Divesh & Suresh

data anonymization using randomization24
Data Anonymization Using Randomization

X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

Information Theory for Data Management - Divesh & Suresh

data anonymization using randomization25
Data Anonymization Using Randomization

X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

Information Theory for Data Management - Divesh & Suresh

data anonymization using randomization26
Data Anonymization Using Randomization

X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

Information Theory for Data Management - Divesh & Suresh

reconstruction of original data distribution
Reconstruction of Original Data Distribution

X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

Reconstruct distribution of X using knowledge of R1 and μ

EM algorithm converges to MLE of original distribution [AA01]

Information Theory for Data Management - Divesh & Suresh

analysis of privacy as00
Analysis of Privacy [AS00]

X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

If X is uniform in [0, 10], privacy determined by range of μ

Information Theory for Data Management - Divesh & Suresh

analysis of privacy aa01
Analysis of Privacy [AA01]

X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

If X is uniform in [0, 1]  [5, 6], privacy smaller than range of μ

Information Theory for Data Management - Divesh & Suresh

analysis of privacy aa0130
Analysis of Privacy [AA01]

X in [0, 10], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

If X is uniform in [0, 1]  [5, 6], privacy smaller than range of μ

In some cases, sensitive value revealed

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy aa01
Quantify Loss of Privacy [AA01]

Goal: quantify loss of privacy based on mutual information I(X;R)

Smaller H(X|R)  more loss of privacy in X by knowledge of R

Larger I(X;R)  more loss of privacy in X by knowledge of R

I(X;R) = H(X) – H(X|R)

I(X;R)used to capture correlation between X and R

p(X) is the prior knowledge of sensitive attribute X

p(X, R) is the joint distribution of X and R

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy aa0132
Quantify Loss of Privacy [AA01]

Goal: quantify loss of privacy based on mutual information I(X;R)

X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy aa0133
Quantify Loss of Privacy [AA01]

Goal: quantify loss of privacy based on mutual information I(X;R)

X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy aa0134
Quantify Loss of Privacy [AA01]

Goal: quantify loss of privacy based on mutual information I(X;R)

X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy aa0135
Quantify Loss of Privacy [AA01]

Goal: quantify loss of privacy based on mutual information I(X;R)

X is uniform in [5, 6], R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

I(X;R) = 0.33

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy aa0136
Quantify Loss of Privacy [AA01]

Goal: quantify loss of privacy based on mutual information I(X;R)

X is uniform in [5, 6], R2 = X + μ (mod 11), μ is uniform in {0, 1}

I(X;R1) = 0.33, I(X;R2) = 0.5  R2 is a bigger privacy risk than R1

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy aa0137
Quantify Loss of Privacy [AA01]

Equivalent goal: quantify loss of privacy based on H(X|R)

X is uniform in [5, 6], R2 = X + μ (mod 11), μ is uniform in {0, 1}

Intuition: we know more about X given R2, than about X given R1

H(X|R1) = 0.67, H(X|R2) = 0.5  R2 is a bigger privacy risk than R1

Information Theory for Data Management - Divesh & Suresh

quantify loss of privacy
Quantify Loss of Privacy

Example: X is uniform in [0, 1]

R3 = e (p = 0.9999), R3 = X (p = 0.0001)

R4 = X (p = 0.6), R4 = 1 – X (p = 0.4)

Is R3 or R4 a bigger privacy risk?

Information Theory for Data Management - Divesh & Suresh

worst case loss of privacy egs03
Worst Case Loss of Privacy [EGS03]

Example: X is uniform in [0, 1]

R3 = e (p = 0.9999), R3 = X (p = 0.0001)

R4 = X (p = 0.6), R4 = 1 – X (p = 0.4)

I(X;R3) = 0.0001 << I(X;R4) = 0.028

Information Theory for Data Management - Divesh & Suresh

worst case loss of privacy egs0340
Worst Case Loss of Privacy [EGS03]

Example: X is uniform in [0, 1]

R3 = e (p = 0.9999), R3 = X (p = 0.0001)

R4 = X (p = 0.6), R4 = 1 – X (p = 0.4)

I(X;R3) = 0.0001 << I(X;R4) = 0.028

But R3 has a larger worst case risk

Information Theory for Data Management - Divesh & Suresh

worst case loss of privacy egs0341
Worst Case Loss of Privacy [EGS03]

Goal: quantify worst case loss of privacy in X by knowledge of R

Use max KL divergence, instead of mutual information

Mutual information can be formulated as expected KL divergence

I(X;R) = ∑x ∑r p(x,r)*log2(p(x,r)/p(x)*p(r)) = KL(p(X,R) || p(X)*p(R))

I(X;R) = ∑r p(r) ∑x p(x|r)*log2(p(x|r)/p(x)) = ER [KL(p(X|r) || p(X))]

[AA01] measure quantifies expected loss of privacy over R

[EGS03] propose a measure based on worst case loss of privacy

IW(X;R) = MAXR [KL(p(X|r) || p(X))]

Information Theory for Data Management - Divesh & Suresh

worst case loss of privacy egs0342
Worst Case Loss of Privacy [EGS03]

Example: X is uniform in [0, 1]

R3 = e (p = 0.9999), R3 = X (p = 0.0001)

R4 = X (p = 0.6), R4 = 1 – X (p = 0.4)

IW(X;R3) = max{0.0, 1.0, 1.0} > IW(X;R4) = max{0.028, 0.028}

Information Theory for Data Management - Divesh & Suresh

worst case loss of privacy egs0343
Worst Case Loss of Privacy [EGS03]

Example: X is uniform in [5, 6]

R1 = X + μ (mod 11), μ is uniform in {-1, 0, 1}

R2 = X + μ (mod 11), μ is uniform in {0, 1}

IW(X;R1) = max{1.0, 0.0, 0.0, 1.0} = IW(X;R2) = {1.0, 0.0, 1.0}

Unable to capture that R2 is a bigger privacy risk than R1

Information Theory for Data Management - Divesh & Suresh

data anonymization summary
Data Anonymization: Summary

Randomization techniques useful for microdata anonymization

Randomization techniques differ in their loss of privacy

Information theoretic measures useful to capture loss of privacy

Expected KL divergence captures expected privacy loss [AA01]

Maximum KL divergence captures worst case privacy loss [EGS03]

Both are useful in practice

Information Theory for Data Management - Divesh & Suresh

outline45
Outline

Part 1

Introduction to Information Theory

Application: Data Anonymization

Application: Database Design

Part 2

Review of Information Theory Basics

Application: Data Integration

Computing Information Theoretic Primitives

Open Problems

Information Theory for Data Management - Divesh & Suresh

information dependencies dr00
Information Dependencies [DR00]

Goal: use information theory to examine and reason about information content of the attributes in a relation instance

Key ideas:

Novel InD measure between attribute sets X, Y based on H(Y|X)

Identify numeric inequalities between InD measures

Results:

InD measures are a broader class than FDs and MVDs

Armstrong axioms for FDs derivable from InD inequalities

MVD inference rules derivable from InD inequalities

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0047
Information Dependencies [DR00]

Functional dependency: X → Y

FD X → Y holds iff  t1, t2 ((t1[X] = t2[X])  (t1[Y] = t2[Y]))

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0048
Information Dependencies [DR00]

Functional dependency: X → Y

FD X → Y holds iff  t1, t2 ((t1[X] = t2[X])  (t1[Y] = t2[Y]))

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0049
Information Dependencies [DR00]

Result: FD X → Y holds iff H(Y|X) = 0

Intuition: once X is known, no remaining uncertainty in Y

H(Y|X) = 0.5

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0050
Information Dependencies [DR00]

Multi-valued dependency: X →→ Y

MVD X →→ Y holds iff R(X,Y,Z) = R(X,Y) R(X,Z)

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0051
Information Dependencies [DR00]

Multi-valued dependency: X →→ Y

MVD X →→ Y holds iff R(X,Y,Z) = R(X,Y) R(X,Z)

=

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0052
Information Dependencies [DR00]

Multi-valued dependency: X →→ Y

MVD X →→ Y holds iff R(X,Y,Z) = R(X,Y) R(X,Z)

=

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0053
Information Dependencies [DR00]

Result: MVD X →→ Y holds iff H(Y,Z|X) = H(Y|X) + H(Z|X)

Intuition: once X known, uncertainties in Y and Z are independent

H(Y|X) = 0.5, H(Z|X) = 0.75, H(Y,Z|X) = 1.25

=

Information Theory for Data Management - Divesh & Suresh

information dependencies dr0054
Information Dependencies [DR00]

Result: Armstrong axioms for FDs derivable from InD inequalities

Reflexivity: If Y  X, then X → Y

H(Y|X) = 0 for Y  X

Augmentation: X → Y  X,Z → Y,Z

0 ≤ H(Y,Z|X,Z) = H(Y|X,Z) ≤ H(Y|X) = 0

Transitivity: X → Y & Y → Z  X → Z

0 ≥ H(Y|X) + H(Z|Y) ≥ H(Z|X) ≥ 0

Information Theory for Data Management - Divesh & Suresh

database normal forms
Database Normal Forms

Goal: eliminate update anomalies by good database design

Need to know the integrity constraints on all database instances

Boyce-Codd normal form:

Input: a set ∑ of functional dependencies

For every (non-trivial) FD R.X → R.Y  ∑+, R.X is a key of R

4NF:

Input: a set ∑ of functional and multi-valued dependencies

For every (non-trivial) MVD R.X →→ R.Y  ∑+, R.X is a key of R

Information Theory for Data Management - Divesh & Suresh

database normal forms56
Database Normal Forms

Functional dependency: X → Y

Which design is better?

=

Information Theory for Data Management - Divesh & Suresh

database normal forms57
Database Normal Forms

Functional dependency: X → Y

Which design is better?

Decomposition is in BCNF

=

Information Theory for Data Management - Divesh & Suresh

database normal forms58
Database Normal Forms

Multi-valued dependency: X →→ Y

Which design is better?

=

Information Theory for Data Management - Divesh & Suresh

database normal forms59
Database Normal Forms

Multi-valued dependency: X →→ Y

Which design is better?

Decomposition is in 4NF

=

Information Theory for Data Management - Divesh & Suresh

well designed databases al03
Well-Designed Databases [AL03]

Goal: use information theory to characterize “goodness” of a database design and reason about normalization algorithms

Key idea:

Information content measure of cell in a DB instance w.r.t. ICs

Redundancy reduces information content measure of cells

Results:

Well-designed DB  each cell has information content > 0

Normalization algorithms never decrease information content

Information Theory for Data Management - Divesh & Suresh

well designed databases al0361
Well-Designed Databases [AL03]

Information content of cell c in database D satisfying FD X → Y

Uniform distribution p(V) on values for c consistent with D\c and FD

Information content of cell c is entropy H(V)

H(V62) = 2.0

Information Theory for Data Management - Divesh & Suresh

well designed databases al0362
Well-Designed Databases [AL03]

Information content of cell c in database D satisfying FD X → Y

Uniform distribution p(V) on values for c consistent with D\c and FD

Information content of cell c is entropy H(V)

H(V22) = 0.0

Information Theory for Data Management - Divesh & Suresh

well designed databases al0363
Well-Designed Databases [AL03]

Information content of cell c in database D satisfying FD X → Y

Information content of cell c is entropy H(V)

Schema S is in BCNF iff  D  S, H(V) > 0, for all cells c in D

Technicalities w.r.t. size of active domain

Information Theory for Data Management - Divesh & Suresh

well designed databases al0364
Well-Designed Databases [AL03]

Information content of cell c in database D satisfying FD X → Y

Information content of cell c is entropy H(V)

H(V12) = 2.0, H(V42) = 2.0

Information Theory for Data Management - Divesh & Suresh

well designed databases al0365
Well-Designed Databases [AL03]

Information content of cell c in database D satisfying FD X → Y

Information content of cell c is entropy H(V)

Schema S is in BCNF iff  D  S, H(V) > 0, for all cells c in D

Information Theory for Data Management - Divesh & Suresh

well designed databases al0366
Well-Designed Databases [AL03]

Information content of cell c in DB D satisfying MVD X →→ Y

Information content of cell c is entropy H(V)

H(V52) = 0.0, H(V53) = 2.32

Information Theory for Data Management - Divesh & Suresh

well designed databases al0367
Well-Designed Databases [AL03]

Information content of cell c in DB D satisfying MVD X →→ Y

Information content of cell c is entropy H(V)

Schema S is in 4NF iff D  S, H(V) > 0, for all cells c in D

Information Theory for Data Management - Divesh & Suresh

well designed databases al0368
Well-Designed Databases [AL03]

Information content of cell c in DB D satisfying MVD X →→ Y

Information content of cell c is entropy H(V)

H(V32) = 1.58, H(V34) = 2.32

Information Theory for Data Management - Divesh & Suresh

well designed databases al0369
Well-Designed Databases [AL03]

Information content of cell c in DB D satisfying MVD X →→ Y

Information content of cell c is entropy H(V)

Schema S is in 4NF iff D  S, H(V) > 0, for all cells c in D

Information Theory for Data Management - Divesh & Suresh

well designed databases al0370
Well-Designed Databases [AL03]

Normalization algorithms never decrease information content

Information content of cell c is entropy H(V)

Information Theory for Data Management - Divesh & Suresh

well designed databases al0371
Well-Designed Databases [AL03]

Normalization algorithms never decrease information content

Information content of cell c is entropy H(V)

=

Information Theory for Data Management - Divesh & Suresh

well designed databases al0372
Well-Designed Databases [AL03]

Normalization algorithms never decrease information content

Information content of cell c is entropy H(V)

=

Information Theory for Data Management - Divesh & Suresh

database design summary
Database Design: Summary

Good database design essential for preserving data integrity

Information theoretic measures useful for integrity constraints

FD X → Y holds iffInD measure H(Y|X) = 0

MVD X →→ Y holds iff H(Y,Z|X) = H(Y|X) + H(Z|X)

Information theory to model correlations in specific database

Information theoretic measures useful for normal forms

Schema S is in BCNF/4NF iff D  S, H(V) > 0, for all cells c in D

Information theory to model distributions over possible databases

Information Theory for Data Management - Divesh & Suresh

outline74
Outline

Part 1

Introduction to Information Theory

Application: Data Anonymization

Application: Database Design

Part 2

Review of Information Theory Basics

Application: Data Integration

Computing Information Theoretic Primitives

Open Problems

Information Theory for Data Management - Divesh & Suresh

review of information theory basics
Review of Information Theory Basics

Discrete distribution: probability p(X)

p(X,Y) = ∑z p(X,Y,Z=z)

Information Theory for Data Management - Divesh & Suresh

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Review of Information Theory Basics

Discrete distribution: probability p(X)

p(Y) = ∑x p(X=x,Y) = ∑x ∑z p(X=x,Y,Z=z)

Information Theory for Data Management - Divesh & Suresh

review of information theory basics77
Review of Information Theory Basics

Discrete distribution: conditional probability p(X|Y)

p(X,Y) = p(X|Y)*p(Y) = p(Y|X)*p(X)

Information Theory for Data Management - Divesh & Suresh

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Review of Information Theory Basics

Discrete distribution: entropy H(X)

h(x) = log2(1/p(x))

H(X) = ∑X=x p(x)*h(x) = 1.75

H(Y) = ∑Y=y p(y)*h(y) = 1.5 (≤ log2(|Y|) = 1.58)

H(X,Y) = ∑X=x ∑Y=y p(x,y)*h(x,y) = 2.25 (≤ log2(|X,Y|) = 2.32)

Information Theory for Data Management - Divesh & Suresh

review of information theory basics79
Review of Information Theory Basics

Discrete distribution: conditional entropy H(X|Y)

h(x|y) = log2(1/p(x|y))

H(X|Y) = ∑X=x ∑Y=y p(x,y)*h(x|y) = 0.75

H(X|Y) = H(X,Y) – H(Y) = 2.25 – 1.5

Information Theory for Data Management - Divesh & Suresh

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Review of Information Theory Basics

Discrete distribution: mutual information I(X;Y)

i(x;y) = log2(p(x,y)/p(x)*p(y))

I(X;Y) = ∑X=x ∑Y=y p(x,y)*i(x;y) = 1.0

I(X;Y) = H(X) + H(Y) – H(X,Y) = 1.75 + 1.5 – 2.25

Information Theory for Data Management - Divesh & Suresh

outline81
Outline

Part 1

Introduction to Information Theory

Application: Data Anonymization

Application: Database Design

Part 2

Review of Information Theory Basics

Application: Data Integration

Computing Information Theoretic Primitives

Open Problems

Information Theory for Data Management - Divesh & Suresh

schema matching
Schema Matching

Goal: align columns across database tables to be integrated

Fundamental problem in database integration

Early useful approach: textual similarity of column names

False positives: Address ≠ IP_Address

False negatives: Customer_Id = Client_Number

Early useful approach: overlap of values in columns, e.g., Jaccard

False positives: Emp_Id ≠ Project_Id

False negatives: Emp_Id = Personnel_Number

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn03
Opaque Schema Matching [KN03]

Goal: align columns when column names, data values are opaque

Databases belong to different government bureaucracies 

Treat column names and data values as uninterpreted (generic)

Example: EMP_PROJ(Emp_Id, Proj_Id, Task_Id, Status_Id)

Likely that all Id fields are from the same domain

Different databases may have different column names

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0384
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

Perform graph matching between GD1 and GD2, minimizing distance

Intuition:

Entropy H(X)captures distribution of values in database column X

Mutual information I(X;Y) captures correlations between X, Y

Efficiency: graph matching between schema-sized graphs

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0385
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0386
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

H(A) = 1.5, H(B) = 2.0, H(C) = 1.0, H(D) = 1.5

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0387
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

H(A) = 1.5, H(B) = 2.0, H(C) = 1.0, H(D) = 1.5, I(A;B) = 1.5

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0388
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

1.5

1.5

A

B

2.0

1.0

1.0

1.5

1.0

1.0

C

D

1.5

0.5

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0389
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

Perform graph matching between GD1 and GD2, minimizing distance

[KN03] uses euclidean and normal distance metrics

1.5

1.5

1.5

2.0

W

A

B

X

2.0

1.5

1.0

1.5

1.0

1.0

1.0

1.5

0.5

1.0

1.0

1.0

C

Y

Z

D

1.5

1.5

0.5

1.0

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0390
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

Perform graph matching between GD1 and GD2, minimizing distance

1.5

1.5

1.5

2.0

W

A

B

X

2.0

1.5

1.0

1.5

1.0

1.0

1.0

1.5

0.5

1.0

1.0

1.0

C

Y

Z

D

1.5

1.5

0.5

1.0

Information Theory for Data Management - Divesh & Suresh

opaque schema matching kn0391
Opaque Schema Matching [KN03]

Approach: build complete, labeled graph GD for each database D

Nodes are columns, label(node(X)) = H(X), label(edge(X, Y)) = I(X;Y)

Perform graph matching between GD1 and GD2, minimizing distance

1.5

1.5

1.5

2.0

W

A

B

X

2.0

1.5

1.0

1.5

1.0

1.0

1.0

1.5

0.5

1.0

1.0

1.0

C

Y

Z

D

1.5

1.5

0.5

1.0

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06
Heterogeneity Identification [DKOSV06]

Goal: identify columns with semantically heterogeneous values

Can arise due to opaque schema matching [KN03]

Key ideas:

Heterogeneity based on distribution, distinguishability of values

Use Information Theory to quantify heterogeneity

Issues:

Which information theoretic measure characterizes heterogeneity?

How do we actually cluster the data ?

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv0693
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv0694
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv0695
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

More semantic types in column  greater heterogeneity

Only email versus email + phone

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv0696
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv0697
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

Relative distribution of semantic types impacts heterogeneity

Mainly email + few phone versus balanced email + phone

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv0698
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv0699
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06100
Heterogeneity Identification [DKOSV06]

Example: semantically homogeneous, heterogeneous columns

More easily distinguished types  greater heterogeneity

Phone + (possibly) SSN versus balanced email + phone

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06101
Heterogeneity Identification [DKOSV06]

Heterogeneity = complexity of describing the data

More, balanced clusters  greater heterogeneity

More distinguishable clusters  greater heterogeneity

Use two views of mutual information

It quantifies the complexity of describing the data (compression)

It quantifies the quality of the compression

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06102
Heterogeneity Identification [DKOSV06]

Hard clustering

Information Theory for Data Management - Divesh & Suresh

measuring complexity of clustering
Measuring complexity of clustering

Take 1: complexity of a clustering = #clusters

standard model of complexity.

Doesn’t capture the fact that clusters have different sizes.

Information Theory for Data Management - Divesh & Suresh

measuring complexity of clustering104
Measuring complexity of clustering

Take 2: Complexity of clustering = number of bits needed to describe it.

Writing down “k” needs log k bits.

In general, let cluster t  T have |t| elements.

set p(t) = |t|/n

#bits to write down cluster sizes = H(T) = S pt log 1/pt

H( ) <

H( )

Information Theory for Data Management - Divesh & Suresh

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Heterogeneity Identification [DKOSV06]

Soft clustering: cluster membership probabilities

How to compute a good soft clustering?

Information Theory for Data Management - Divesh & Suresh

measuring complexity of clustering106
Measuring complexity of clustering

Take 1:

p(t) = Sx p(x) p(t|x)

Compute H(T) as before.

Problem:

H(T1) = H(T2) !!

Information Theory for Data Management - Divesh & Suresh

measuring complexity of clustering107
Measuring complexity of clustering

By averaging the memberships, we’ve lost useful information.

Take II: Compute I(T;X) !

Even better: If T is a hard clustering of X, then I(T;X) = H(T)

I(T2;X) = 0.46

I(T1;X) = 0

Information Theory for Data Management - Divesh & Suresh

measuring cost of a cluster
Measuring cost of a cluster

Given objects Xt = {X1, X2, …, Xm} in cluster t,

Cost(t) = sum of distances from Xi to cluster center

If we set distance to be KL-distance, then

Cost of a cluster = I(Xt,V)

Information Theory for Data Management - Divesh & Suresh

cost of a clustering
Cost of a clustering

If we partition X into k clusters X1, ..., Xk

Cost(clustering) = Si pi I(Xi, V) (pi = |Xi|/|X|)

Suppose we treat each cluster center itself as a point ?

Information Theory for Data Management - Divesh & Suresh

cost of a clustering110
Cost of a clustering

We can write down the “cost” of this “cluster”

Cost(T) = I(T;V)

Key result [BMDG05] :

Cost(clustering) = I(X, V) – I(T, V)

Minimizing cost(clustering) => maximizing I(T, V)

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06111
Heterogeneity Identification [DKOSV06]

Represent strings as q-gram distributions

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06112
Heterogeneity Identification [DKOSV06]

iIB: find soft clustering T of X that minimizes I(T;X) – β*I(T;V)

Allow iIB to use arbitrarily many clusters, use β* = H(X)/I(X;V)

Closest to point with minimum space and maximum quality

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06113
Heterogeneity Identification [DKOSV06]

Rate distortion curve: I(T;V)/I(X;V) vs I(T;X)/H(X)

β*

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06114
Heterogeneity Identification [DKOSV06]

Heterogeneity = mutual information I(T;X) of iIB clustering T at β*

0 ≤I(T;X) (= 0.126) ≤ H(X) (= 2.0), H(T) (= 1.0)

Ideally use iIB with an arbitrarily large number of clusters in T

Information Theory for Data Management - Divesh & Suresh

heterogeneity identification dkosv06115
Heterogeneity Identification [DKOSV06]

Heterogeneity = mutual information I(T;X) of iIB clustering T at β*

Information Theory for Data Management - Divesh & Suresh

data integration summary
Data Integration: Summary

Analyzing database instance critical for effective data integration

Matching and quality assessments are key components

Information theoretic measures useful for schema matching

Align columns when column names, data values are opaque

Mutual information I(X;V) captures correlations between X, V

Information theoretic measures useful for heterogeneity testing

Identify columns with semantically heterogeneous values

I(T;X) of iIB clustering T at β* captures column heterogeneity

Information Theory for Data Management - Divesh & Suresh

outline117
Outline

Part 1

Introduction to Information Theory

Application: Data Anonymization

Application: Database Design

Part 2

Review of Information Theory Basics

Application: Data Integration

Computing Information Theoretic Primitives

Open Problems

Information Theory for Data Management - Divesh & Suresh

domain size matters
Domain size matters
  • For random variable X, domain size = supp(X) = {xi | p(X = xi) > 0}
  • Different solutions exist depending on whether domain size is “small” or “large”
  • Probability vectors usually very sparse

Information Theory for Data Management - Divesh & Suresh

entropy case i small domain size
Entropy: Case I - Small domain size
  • Suppose the #unique values for a random variable X is small (i.e fits in memory)
  • Maximum likelihood estimator:
    • p(x) = #times x is encountered/total number of items in set.

1

2

1

2

1

5

4

1

2

3

4

5

Information Theory for Data Management - Divesh & Suresh

entropy case i small domain size120
Entropy: Case I - Small domain size
  • HMLE = Sx p(x) log 1/p(x)
  • This is a biased estimate:
    • E[HMLE] < H
  • Miller-Madow correction:
    • H’ = HMLE + (m’ – 1)/2n
      • m’ is an estimate of number of non-empty bins
      • n = number of samples
  • Bad news: ALL estimators for H are biased.
  • Good news: we can quantify bias and variance of MLE:
    • Bias <= log(1 + m/N)
    • Var(HMLE) <= (log n)2/N

Information Theory for Data Management - Divesh & Suresh

entropy case ii large domain size
Entropy: Case II - Large domain size
  • |X| is too large to fit in main memory, so we can’t maintain explicit counts.
  • Streaming algorithms for H(X):
    • Long history of work on this problem
    • Bottomline:

(1+e)-relative-approximation for H(X) that allows for updates to frequencies, and requires “almost constant”, and optimal space [HNO08].

Information Theory for Data Management - Divesh & Suresh

streaming entropy ccm07
Streaming Entropy [CCM07]
  • High level idea: sample randomly from the stream, and track counts of elements picked [AMS]
  • PROBLEM: skewed distribution prevents us from sampling lower-frequency elements (and entropy is small)
  • Idea: estimate largest frequency, and

distribution of what’s left (higher entropy)

Information Theory for Data Management - Divesh & Suresh

streaming entropy ccm07123
Streaming Entropy [CCM07]
  • Maintain set of samples from original distribution and distribution without most frequent element.
  • In parallel, maintain estimator for frequency of most frequent element
    • normally this is hard
    • but if frequency is very large, then simple estimator exists [MG81] (Google interview puzzle!)
  • At the end, compute function of these two estimates
  • Memory usage: roughly 1/e2 log(1/e) (e is the error)

Information Theory for Data Management - Divesh & Suresh

entropy and mi are related
Entropy and MI are related
  • I(X;Y) = H(X,Y) – H(X) – H(Y)
  • Suppose we can c-approximate H(X) for any c > 0:

Find H’(X) s.t |H(X) – H’(X)| <= c

  • Then we can 3c-approximate I(X;Y):
    • I(X;Y) = H(X,Y) – H(X) – H(Y)

<= H’(X,Y)+c – (H’(X)-c) – (H’(Y)-c)

<= H’(X,Y) – H’(X) – H’(Y) + 3c

<= I’(X,Y) + 3c

  • Similarly, we can 2c-approximate H(Y|X) = H(X,Y) – H(X)
  • Estimating entropy allows us to estimate I(X;Y) and H(Y|X)

Information Theory for Data Management - Divesh & Suresh

computing kl divergence small domains
Computing KL-divergence: Small Domains
  • “easy algorithm”: maintain counts for each of p and q, normalize, and compute KL-divergence.
  • PROBLEM ! Suppose qi = 0:
    • pi log pi/qi is undefined !
  • General problem with ML estimators: all events not seen have probability zero !!
    • Laplace correction: add one to counts for each seen element
    • Slightly better: add 0.5 to counts for each seen element [KT81]
    • Even better, more involved: use Good-Turing estimator [GT53]
  • YIeld non-zero probability for “things not seen”.

Information Theory for Data Management - Divesh & Suresh

computing kl divergence large domains
Computing KL-divergence: Large Domains
  • Bad news: No good relative-approximations exist in small space.
  • (Partial) good news: additive approximations in small space under certain technical conditions (no pi is too small).
  • (Partial) good news: additive approximations for symmetric variant of KL-divergence, via sampling.
  • For details, see [GMV08,GIM08]

Information Theory for Data Management - Divesh & Suresh

information theoretic clustering
Information-theoretic Clustering
  • Given a collection of random variables X, each “explained” by a random variable Y, we wish to find a (hard or soft) clustering T such that

I(T,X) – bI(T, Y)

is minimized.

  • Features of solutions thus far:
    • heuristic (general problem is NP-hard)
    • address both small-domain and large-domain scenarios.

Information Theory for Data Management - Divesh & Suresh

agglomerative clustering aib st00
Agglomerative Clustering (aIB) [ST00]
  • Fix number of clusters k
  • While number of clusters < k
    • Determine two clusters whose merge loses the least information
    • Combine these two clusters
  • Output clustering
  • Merge Criterion:
    • merge the two clusters so that change in I(T;V) is minimized
  • Note: no consideration of b (number of clusters is fixed)

Information Theory for Data Management - Divesh & Suresh

agglomerative clustering aib s
Agglomerative Clustering (aIB) [S]
  • Elegant way of finding the two clusters to be merged:
  • Let dJS(p,q) = (1/2)(dKL(p,m) + dKL(q,m)), m = (p+q)/2
  • dJS(p,q) is a symmetric distance between p, q (Jensen-Shannon distance)
  • We merge clusters that have smallest dJS(p,q), (weighted by cluster mass)

p

m

q

Information Theory for Data Management - Divesh & Suresh

iterative information bottleneck iib s
Iterative Information Bottleneck (iIB) [S]
  • aIB yields a hard clustering with k clusters.
  • If you want a soft clustering, use iIB (variant of EM)
    • Step 1: p(t|x) ← exp(-bdKL(p(V|x),p(V|t))
      • assign elements to clusters in proportion (exponentially) to distance from cluster center !
    • Step 2: Compute new cluster centers by computing weighted centroids:
      • p(t) = Sx p(t|x) p(x)
      • p(V|t) = Sx p(V|t) p(t|x) p(x)/p(t)
    • Choose b according to [DKOSV06]

Information Theory for Data Management - Divesh & Suresh

dealing with massive data sets
Dealing with massive data sets
  • Clustering on massive data sets is a problem
  • Two main heuristics:
    • Sampling [DKOSV06]:
      • pick a small sample of the data, cluster it, and (if necessary) assign remaining points to clusters using soft assignment.
      • How many points to sample to get good bounds ?
    • Streaming:
      • Scan the data in one pass, performing clustering on the fly
      • How much memory needed to get reasonable quality solution ?

Information Theory for Data Management - Divesh & Suresh

limbo for aib atms04
LIMBO (for aIB) [ATMS04]
  • BIRCH-like idea:
    • Maintain (sparse) summary for each cluster (p(t), p(V|t))
    • As data streams in, build clusters on groups of objects
    • Build next-level clusters on cluster summaries from lower level

Information Theory for Data Management - Divesh & Suresh

outline133
Outline

Part 1

Introduction to Information Theory

Application: Data Anonymization

Application: Database Design

Part 2

Review of Information Theory Basics

Application: Data Integration

Computing Information Theoretic Primitives

Open Problems

Information Theory for Data Management - Divesh & Suresh

open problems
Open Problems
  • Data exploration and mining – information theory as first-pass filter
  • Relation to nonparametric generative models in machine learning (LDA, PPCA, ...)
  • Engineering and stability: finding right knobs to make systems reliable and scalable
  • Other information-theoretic concepts ? (rate distortion, higher-order entropy, ...)

THANK YOU !

Information Theory for Data Management - Divesh & Suresh

references information theory
References: Information Theory

[CT] Tom Cover and Joy Thomas: Information Theory.

[BMDG05] Arindam Banerjee, Srujana Merugu, Inderjit Dhillon, Joydeep Ghosh. Learning with Bregman Divergences, JMLR 2005.

[TPB98] Naftali Tishby, Fernando Pereira, William Bialek. The Information Bottleneck Method. Proc. 37th Annual Allerton Conference, 1998.

Information Theory for Data Management - Divesh & Suresh

references data anonymization
References: Data Anonymization

[AA01] Dakshi Agrawal, Charu C. Aggarwal: On the design and quantification of privacy preserving data mining algorithms. PODS 2001.

[AS00] Rakesh Agrawal, Ramakrishnan Srikant: Privacy preserving data mining. SIGMOD 2000.

[EGS03] Alexandre Evfimievski, Johannes Gehrke, Ramakrishnan Srikant: Limiting privacy breaches in privacy preserving data mining. PODS 2003.

Information Theory for Data Management - Divesh & Suresh

references database design
References: Database Design

[AL03] Marcelo Arenas, Leonid Libkin: An information theoretic approach to normal forms for relational and XML data. PODS 2003.

[AL05] Marcelo Arenas, Leonid Libkin: An information theoretic approach to normal forms for relational and XML data. JACM 52(2), 246-283, 2005.

[DR00] Mehmet M. Dalkilic, Edward L. Robertson: Information dependencies. PODS 2000.

[KL06] Solmaz Kolahi, Leonid Libkin: On redundancy vs dependency preservation in normalization: an information-theoretic study of XML. PODS 2006.

Information Theory for Data Management - Divesh & Suresh

references data integration
References: Data Integration

[AMT04] Periklis Andritsos, Renee J. Miller, Panayiotis Tsaparas: Information-theoretic tools for mining database structure from large data sets. SIGMOD 2004.

[DKOSV06] Bing Tian Dai, Nick Koudas, Beng Chin Ooi, Divesh Srivastava, Suresh Venkatasubramanian: Rapid identification of column heterogeneity. ICDM 2006.

[DKSTV08] Bing Tian Dai, Nick Koudas, Divesh Srivastava, Anthony K. H. Tung, Suresh Venkatasubramanian: Validating multi-column schema matchings by type. ICDE 2008.

[KN03] Jaewoo Kang, Jeffrey F. Naughton: On schema matching with opaque column names and data values. SIGMOD 2003.

[PPH05] Patrick Pantel, Andrew Philpot, Eduard Hovy: An information theoretic model for database alignment. SSDBM 2005.

Information Theory for Data Management - Divesh & Suresh

references computing it quantities
References: Computing IT quantities

[P03] Liam Panninski. Estimation of entropy and mutual information. Neural Computation 15: 1191-1254.

[GT53] I. J. Good. Turing’s anticipation of Empirical Bayes in connection with the cryptanalysis of the Naval Enigma. Journal of Statistical Computation and Simulation, 66(2), 2000.

[KT81] R. E. Krichevsky and V. K. Trofimov. The performance of universal encoding. IEEE Trans. Inform. Th. 27 (1981), 199--207.

[CCM07] Amit Chakrabarti, Graham Cormode and Andrew McGregor. A near-optimal algorithm for computing the entropy of a stream. Proc. SODA 2007.

Information Theory for Data Management - Divesh & Suresh

references computing it quantities140
References: Computing IT quantities

[HNO] Nich Harvey, Jelani Nelson, Krzysztof Onak. Sketching and Streaming Entropy via Approximation Theory. FOCS 2008.

[ATMS04] Periklis Andritsos, Panayiotis Tsaparas, Renée J. Miller and Kenneth C. Sevcik. LIMBO: Scalable Clustering of Categorical Data. EDBT 2004.

[S] Noam Slonim. The Information Bottleneck: theory and applications. Ph.D Thesis. Hebrew University, 2000.

[GMV08] Sudipto Guha, Andrew McGregor, Suresh Venkatasubramanian. Streaming and sublinear approximations for information distances. ACM Trans Alg. 2008.

[GIM08] Sudipto Guha, Piotr Indyk, Andrew McGregor. Sketching Information Distances. JMLR, 2008.

Information Theory for Data Management - Divesh & Suresh

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