Bee-Identification Error Exponent with Absentee Bees Anshoo Tandon - - PowerPoint PPT Presentation

Bee-Identification Error Exponent with Absentee Bees

Anshoo Tandon

National University of Singapore

Joint work with: Vincent Y. F. Tan, NUS Lav R. Varshney, UIUC ISIT 2020

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 1 / 18

Bee-Identification Problem

To identify bees moving on a beehive Each bee labeled with a unique barcode A camera is employed to picture the beehive Application:

◮ In understanding interactions among honeybees

(T. Gernat et al., PNAS’18)

◮ To study similarity with human social-networks ◮ Model large-scale social networks for studying disease transmission

Finite resolution image adds noise to barcodes

◮ Noise may cause bee-identification error

We pose this as an information-theoretic problem

◮ Represent each barcode as a binary vector of length n ◮ Let m denote the total number of bees Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 2 / 18

Previous Work: All m bees are present in the image

Ref: Tandon, Tan, and Varshney (TCOM’2019)

Model:

◮ Each barcode is represented as a binary codeword of length n ◮ Collect all the m codewords to form a codebook C ⋆ Codebook C has size m × n ⋆ Each barcode corresponds to a row-vector (of length n) in C ◮ Given a beehive image, extract all the barcodes ⋆ Stack the barcodes are in a single column ⋆ The effective channel is as follows:

Effective Channel Codebook C ✲ Row-Permutation π Cπ ✲ BSC(p) ✲

˜ Cπ

◮ The channel permutes the rows of C and then adds noise ⋆ i-th row of Cπ corresponds to the π(i)-th row of C ⋆ π is uniformly distributed over the set of all m-letter permutations Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 3 / 18

Absentee Bees: k out of m bees are absent in the image

Effective Channel Codebook C✲ Row-Permutation π

✲

Cπ Delete k rows

✲

Cπ(m−k)

BSC(p)

✲

˜ Cπ(m−k)

Channel deletes k rows of Cπ, to model the scenario in which k bees,

• ut of a total of m bees, are absent in the beehive image

Without loss of generality, we assume that the channel deletes the last k rows of Cπ to produce Cπ(m−k)

◮ π(m−k): injective mapping of m − k rows of Cπ(m−k) to m rows of C

Noise is modeled as BSC(p), with 0 < p < 0.5 Decoder: has knowledge of codebook C Decoder’s task is to recover the channel-induced mapping π(m−k) using the channel output ˜ Cπ(m−k)

◮ π(m−k) directly ascertains the identity of the m − k bees in the image Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 4 / 18

Channel Model . . .

Effective Channel Codebook C✲ Row-Permutation π

✲

Cπ Delete k rows

✲

Cπ(m−k)

BSC(p)

✲

˜ Cπ(m−k)

For 1 ≤ i ≤ m − k, the i-th row of ˜ Cπ(m−k), denoted ˜ ❝i, is a noisy version of ❝π(i), and we have Pr{˜ ❝i | ❝π(i)} = pdi(1 − p)n−di, 1 ≤ i ≤ m − k, Pr ˜ Cπ(m−k)

• C, π(m−k)
• =

m−k

• i=1

pdi(1 − p)n−di, where di dH(˜ ❝i, ❝π(i)) denotes the Hamming distance between vectors ˜ ❝i and ❝π(i)

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 5 / 18

Bee-Identification: Problem Formulation

Effective Channel Codebook C✲ Row-Permutation π

✲

Cπ Delete k rows

✲

Cπ(m−k)

BSC(p)

✲

˜ Cπ(m−k)

The decoder corresponds to a function φ that takes ˜ Cπ(m−k) as an input and produces a map ν : {1, . . . , m −k} → {1, . . . , m} as output

◮ ν(i) corresponds to the index of the transmitted codeword which

produced the received word ˜ ❝i, where 1 ≤ i ≤ m − k

The indicator variable for the bee-identification error is defined as D

• ν, π(m−k)
• 1,

if ν = π(m−k), 0, if ν = π(m−k). The expected bee-identification error probability over the BSC(p) is D(C, p, k, φ) Eπ(m−k)

• E
• D
• ν, π(m−k)
• ,

◮ inner expectation is over the distribution of ˜

Cπ(m−k) given C and π(m−k)

◮ outer expectation is over the uniform distribution of π(m−k) over the

set of all injective maps from {1, . . . , m − k} to {1, . . . , m}

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 6 / 18

Bee-Identification Error Exponent

Let C (n, m) denote the set of all binary codebooks of size m × n Given n, m, and k, define the minimum expected bee-identification error probability as D(n, m, p, k) min

C,φ D(C, p, k, φ),

where the minimum is over all codebooks C ∈ C (n, m), and all decoding functions φ For a given R > 0 and α ∈ (0, 1), we consider the setting where m = 2nR, and k = αm = α2nR, and study the error exponent ED(R, p, α) lim sup

n→∞ −1

n log D(n, 2nR, p, α2nR)

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 7 / 18

Our Contributions

We provide an exact characterization of the error exponent

◮ We show that independent barcode decoding is optimal, i.e., joint

decoding does not result in a better error exponent relative to the independent decoding of each noisy barcode

◮ This is in contrast to the result without absentee bees, where joint

barcode decoding results in a significantly higher error exponent than independent barcode decoding

We characterize the bee-identification “capacity” (i.e., supremum of rates for which the error probability can be made arbitrarily small)

◮ We prove the strong converse showing that for rates greater than the

capacity, the error probability tends to 1 when n → ∞

We show that for low rates, there is a discontinuity in the error exponent function at α = 0

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 8 / 18

Independent Decoding: Upper Bound on D(n, m, p, k)

Here, the decoder picks ˜ ❝i, and produces the index ν(i) = arg min1≤j≤m dH(˜ ❝i, ❝j), for 1 ≤ i ≤ m − k Using the union bound, we get D(C, p, k, φI) ≤

m−k

• i=1

Eπ(m−k)

• Pr
• ν(i) = π(m−k)(i)
• Now, D(n, m, p, k) can be upper bounded as

D(n, m, p, k) ≤ min {1, (m − k) Pe(n, m, p)} , where Pe(n, m, p) denotes the minimum achievable average error probability over BSC(p), for transmission of a message, using a codebook with m codewords, each having length n

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 9 / 18

Joint Decoding: Lower Bound on D(n, m, p, k)

D(n, m, p, k) can be lower bounded, using joint ML decoding of bee barcodes, as follows D(n, m, p, k) > 1−2ε 2 min

• 1, (m−k)ε Pe(n, ⌊kε⌋, p)
• ,

where 0 < ε < 1/2 and k > 1/ε D(n, m, p, k) can alternatively be lower bounded as follows D(n, m, p, k) > (1 − 2ε)

• 1 − exp (−(m − k)ε Pe(n, ⌊kε⌋, p))
• ◮ These lower bounds are obtained by only considering the error events

where only a single bee barcode is incorrectly decoded

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 10 / 18

Bee-Identification Error Exponent

For R > 0, the reliability function is defined as E(R, p) lim sup

n→∞ −1

n log Pe(n, 2nR, p) The bee-identification error exponent is defined as ED(R, p, α) lim sup

n→∞ −1

n log D(n, 2nR, p, α2nR) Bounds on D(n, m, p, k) can be applied to obtain the following:

Theorem 1

For 0 < α < 1, we have ED(R, p, α) = |E(R, p) − R|+, where |x|+ max(0, x). Further, this exponent is achieved via independent decoding of each barcode.

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 11 / 18

‘Capacity’ of the bee-identification problem

Given 0 ≤ ǫ < 1, define the bee-identification ǫ-capacity as CD(p, α, ǫ)sup

• R : lim inf

n→∞ D(n, 2nR, p, α2nR)≤ǫ

• The capacity is exactly characterized by the following theorem:

Theorem 2

For 0 < α < 1, and every 0 ≤ ǫ < 1, we have CD(p, α, ǫ) = R∗

p,

where R∗

p is unique positive solution of the following equation

E(R, p) = R. The strong converse property holds, i.e., if R > R∗

p, then the error

probability D(n, 2nR, p, α2nR) tends to 1 as n → ∞

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 12 / 18

Error Exponent when α ↓ 0

Let ED(R, p) be the bee-identification error exponent when all bees are present (with k = 0)

Theorem 3

For 0 < R < min {0.169, Rex/2}, we have the following strict inequality lim

α↓0 ED(R, p, α) < ED(R, p).

◮ The above theorem highlights a discontinuity in the bee-identification

error exponent function at α = 0

◮ Contrasting behaviors: ⋆ Independent decoding of bee barcodes is optimal for the absentee bee

scenario, even when arbitrarily small fraction of bees are absent

⋆ When all bees are present, strictly better error exponent is achieved via

joint ML decoding of barcodes

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 13 / 18

Numerical Results: Error Exponent

0.05 0.1 0.15 0.2 0.25 0.3057 0.35 0.1 0.2 0.3 0.4 0.5 0.6

Bounds on ED(R, p, α) for p = 0.05.

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 14 / 18

Numerical Results: Error Exponent

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.5 1 1.5 2 2.5

Bounds on the bee-identification error exponent, when p = 0.01.

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 15 / 18

Numerical Results: Capacity

0.01 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.49 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Lower and upper bounds on CD(p, α, ǫ).

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 16 / 18

Conclusions and Future Work

We provided an exact characterization of the error exponent and capacity for bee-identification problem with absentee bees These results (for BSC(p)) can be extended to general DMCs Future work:

◮ Study the error exponent for the scenario where α, the fraction of

absentee bees, also varies with blocklength n

◮ Obtain second-order results, i.e., the scaling of the code rate for a

given ǫ, and finite n

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 17 / 18

Thank You!

Please contact us, in case you have any queries: Anshoo Tandon (anshoo.tandon@gmail.com) Vincent Y. F. Tan (vtan@nus.edu.sg) Lav R. Varshney (varshney@illinois.edu)

Anshoo Tandon (NUS) Bee-Identification with Absentee Bees ISIT 2020 18 / 18