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Recent advances in DNA sequencing technology and DNA storage systems have rekindled the interest in deletion channels. Multiple recent works have looked at variants of sequence reconstruction over a single and over multiple deletion channels, a notoriously difficult problem due to its highly combinatorial nature. Although works in theoretical computer science have provided algorithms which guarantee perfect reconstruction with multiple independent observations from the deletion channel, they are only applicable in the large blocklength regime and more restrictively, when the number of observations is also large. Indeed, with only a few observations, perfect reconstruction of the input sequence may not even be possible in most cases. In such situations, maximum likelihood (ML) and maximum aposteriori (MAP) estimates for the deletion channels are natural questions that arise and these have remained open to the best of our knowledge. In this work, we take steps to answer the two aforementioned questions. Specifically: 1. We show that solving for the ML estimate over the single deletion channel (which can be cast as a discrete optimization problem) is equivalent to solving its relaxation, a continuous optimization problem; 2. We exactly compute the symbolwise posterior distributions (under some assumptions on the priors) for both the single as well as multiple deletion channels. As part of our contributions, we also introduce tools to visualize and analyze error events, which we believe could be useful in other related problems concerning deletion channels.
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Equivalence of ML decoding to a continuous optimization problem
Maximum likelihood (ML) and symbolwise maximum aposteriori (MAP) estimation for discrete input sequences play a central role in a number of applications that arise in communications, information and coding theory. Many instances of these problems are proven to be intractable, for example through reduction to NP-complete integer optimization problems. In this work, we prove that the ML estimation of a discrete input sequence (with no assumptions on the encoder/channel used) is equivalent to the solution of a continuous non-convex optimization problem, and that this formulation is closely related to the computation of symbolwise MAP estimates. This equivalence is particularly useful in situations where a function we term the expected likelihood is efficiently computable. In such situations, we give a ML heuristic and show numerics for sequence estimation over the deletion channel.
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- PAR ID:
- 10273068
- Date Published:
- Journal Name:
- IEEE International Symposium on Information Theory (ISIT)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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