# LOOP CALCULUS AND BELIEF PROPAGATION FOR Q-ARY ALPHABET: LOOP TOWER

## Abstract

Loop calculus introduced in [1], [2] constitutes a new theoretical tool that explicitly expresses symbol Maximum-A-Posteriori (MAP) solution of a general statistical inference problem via a solution of the Belief Propagation (BP) equations. This finding brought a new significance to the BP concept, which in the past was thought of as just a loop-free approximation. In this paper they continue a discussion of the Loop Calculus, partitioning the results into three Sections. In Section 1 they introduce a new formulation of the Loop Calculus in terms of a set of transformations (gauges) that keeping the partition function of the problem invariant. The full expression contains two terms referred to as the 'ground state' and 'excited states' contributions. The BP equations are interpreted as a special (BP) gauge fixing condition that emerges as a special orthogonality constraint between the ground state and excited states, which also selects loop contributions as the only surviving ones among the excited states. In Section 2 they demonstrate how the invariant interpretation of the Loop Calculus, introduced in Section 1, allows a natural extension to the case of a general q-ary alphabet, this is achieved via a loop tower sequential construction. The ground level in themore »

- Authors:

- Los Alamos National Laboratory

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- OSTI Identifier:
- 1000500

- Report Number(s):
- LA-UR-07-0149

TRN: US201101%%635

- DOE Contract Number:
- AC52-06NA25396

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99; COLOR; CONSTRUCTION; EXCITED STATES; GROUND LEVEL; GROUND STATES; PARTITION FUNCTIONS; TRANSFORMATIONS

### Citation Formats

```
CHERTKOV, MICHAEL, and CHERNYAK, VLADIMIR.
```*LOOP CALCULUS AND BELIEF PROPAGATION FOR Q-ARY ALPHABET: LOOP TOWER*. United States: N. p., 2007.
Web. doi:10.2172/1000500.

```
CHERTKOV, MICHAEL, & CHERNYAK, VLADIMIR.
```*LOOP CALCULUS AND BELIEF PROPAGATION FOR Q-ARY ALPHABET: LOOP TOWER*. United States. doi:10.2172/1000500.

```
CHERTKOV, MICHAEL, and CHERNYAK, VLADIMIR. Wed .
"LOOP CALCULUS AND BELIEF PROPAGATION FOR Q-ARY ALPHABET: LOOP TOWER". United States.
doi:10.2172/1000500. https://www.osti.gov/servlets/purl/1000500.
```

```
@article{osti_1000500,
```

title = {LOOP CALCULUS AND BELIEF PROPAGATION FOR Q-ARY ALPHABET: LOOP TOWER},

author = {CHERTKOV, MICHAEL and CHERNYAK, VLADIMIR},

abstractNote = {Loop calculus introduced in [1], [2] constitutes a new theoretical tool that explicitly expresses symbol Maximum-A-Posteriori (MAP) solution of a general statistical inference problem via a solution of the Belief Propagation (BP) equations. This finding brought a new significance to the BP concept, which in the past was thought of as just a loop-free approximation. In this paper they continue a discussion of the Loop Calculus, partitioning the results into three Sections. In Section 1 they introduce a new formulation of the Loop Calculus in terms of a set of transformations (gauges) that keeping the partition function of the problem invariant. The full expression contains two terms referred to as the 'ground state' and 'excited states' contributions. The BP equations are interpreted as a special (BP) gauge fixing condition that emerges as a special orthogonality constraint between the ground state and excited states, which also selects loop contributions as the only surviving ones among the excited states. In Section 2 they demonstrate how the invariant interpretation of the Loop Calculus, introduced in Section 1, allows a natural extension to the case of a general q-ary alphabet, this is achieved via a loop tower sequential construction. The ground level in the tower is exactly equivalent to assigning one color (out of q available) to the 'ground state' and considering all 'excited' states colored in the remaining (q-1) colors, according to the loop calculus rule. Sequentially, the second level in the tower corresponds to selecting a loop from the previous step, colored in (q-1) colors, and repeating the same ground vs excited states splitting procedure into one and (q-2) colors respectively. The construction proceeds till the full (q-1)-levels deep loop tower (and the corresponding contributions to the partition function) are established. In Section 3 they discuss an ultimate relation between the loop calculus and the Bethe-Free energy variational approach of [3].},

doi = {10.2172/1000500},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Wed Jan 10 00:00:00 EST 2007},

month = {Wed Jan 10 00:00:00 EST 2007}

}