# Feedback-aided complexity reductions in ML and lattice decoding

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TL;DR: Upper bounds on the feedback-aided complexity exponent required for the broad families of ML-based and lattice based decoders to achieve the optimal diversity-multiplexing behavior are derived.

Abstract: The work analyzes the computational-complexity savings that a single bit of feedback can provide in the computationally intense setting of non-ergodic MIMO communications. Specifically we derive upper bounds on the feedback-aided complexity exponent required for the broad families of ML-based and lattice based decoders to achieve the optimal diversity-multiplexing behavior. The bounds reveal a complexity that is reduced from being exponential in the number of codeword bits, to being at most exponential in the rate. Finally the derived savings are met by practically constructed ARQ schemes, as well as simple lattice designs, decoders, and computation-halting policies.

## Summary (2 min read)

### Introduction

- Computational complexity and ratereliability performance are highly intertwined, in the sense that limitations to computational resources (commonly measured by floating point operations - flops), bring about substantial degradation in the system performance.
- In the high rate setting of interest, the lion’s share of computational costs is due to decoding algorithms, on which the authors here focus, specifically considering the broad family of ML-based and regularized (MMSE-preprocessed) lattice decoding algorithms.
- In the high SNR regime (SNR will be henceforth denoted as ρ), this relationship was described in [1] using the high SNR measures of multiplexing gain r := R/ log ρ and diversity gain d(r) := − limρ→∞ logPerr/ log ρ.
- Motivated by the considerable magnitude of the complexity exponent in (2), the authors here seek to understand the role of feedback in reducing complexity, rather than in improving reliability.
- For this the authors seek to quantify the feedback-aided complexity exponent required to achieve the original d∗(r) in the presence of a modified version of the above mentioned Lround MIMO ARQ.

### A. MIMO-ARQ signaling

- The authors here present the general nT×nR MIMO-ARQ signaling setting, and focus on the details which are necessary for their exposition.
- For further understanding of the MIMO-ARQ channel, the reader is referred to [5] as well as [6].
- The authors note that the signals XARQ,LC are drawn from a lattice design that ensures unique decodability at every round6.
- The authors proceed with quantifying the complexity reductions due to ARQ feedback.

### II. COMPLEXITY REDUCTION USING ARQ FEEDBACK

- The authors here seek to analyze the complexity reductions due to MIMO ARQ feedback.
- Specifically for d∗(r) denoting the optimal DMT of the nT × nR MIMO channel in the absence of feedback, the authors here seek to describe the feedback-aided complexity exponent required to meet the same d∗(r) with the assistance now of an L-round ARQ scheme.
- A minimum delay ARQ scheme with L = nT rounds achieves d∗(r) with c(r) ≤ cred(r), irrespective of the ARQ-compatible, minimum delay, NVD, rate-1 lattice design, for any aggressive intermediate halting policy, and any sphere decoding order policy, also known as Proposition 1.
- The proof of this proposition will appear later on, and is crucial in the achievability part of the proof of Theorem 1.
- For general lattice designs derived from cyclic division algebra (CDA) (cf. [8], [9]), F and L are number fields, with L a finite, cyclic Galois extension of F of degree n. Let σ denote a generator of the Galois group Gal(L/F).

### A. Feedback reduction for asymmetric channels: nR ≤ nT

- The authors now consider the case of nR ≤ nT, and specifically the case where nR|nT (i.e., nT is an integer multiple of nR), to observe again how simple implementations offer substantial reductions in complexity.
- In terms of statistics, the results hold for any i.i.d. regular fading distribution.
- The authors see a considerably reduced complexity of the feedback aided scheme (Fig. 2(a), lower line) which, at the same time, achieves a much higher DMT performance (Fig. 2(b), upper line) than its non-feedback counterpart.

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### Cites background from "Feedback-aided complexity reduction..."

...This work improves upon the result of [14] and identifies the first practically constructed feedback schemes, as well as simple lattice code designs and decoders, that jointly guarantee d∗(r) with just a polynomial time complexity....

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...[3], [13]) and non-feasibility of LR-aided methods, the work in [14] showed that if the feedback is used for reducing complexity, rather than in improving reliability as shown in [15], then a properly positioned single bit of feedback can provide exponential reductions in the...

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...Finding out computationally efficient decoding algorithms that allow for near-optimal behavior with reduced complexity cost remains an important research topic of substantial practical interest ([4]–[14])....

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...the case tough that feedback-aided complexity of [14], albeit significantly smaller than those required in the absence of feedback (cf....

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##### References

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### "Feedback-aided complexity reduction..." refers background in this paper

...Under ARQ signaling, each message is associated to a unique block [X1C X 2 C · · ·XLC ] of signaling matrices, where each XiC ∈ CnT×T , i = 1, · · · , L, corresponds to the nT×T matrix of signals sent during the ith round....

[...]

...We here present the general nT×nR MIMO-ARQ signaling setting, and focus on the details which are necessary for our exposition....

[...]

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