Structured spherical codes with asymptotically optimal distance distributions

Taylor, Robert M.; Mili, Lamine; Zaghloul, Amir I.

doi:10.1109/isit.2017.8006917

Cited by 5 publications

(8 citation statements)

References 23 publications

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“…We iterate until the deviation between consecutive cost function evaluations is less than a tolerance level of 10 −3 . Table I shows the optimizing radii vector r, the global circumradius (tube radius) ρG,r, and the tube density from (24) for a small sample of even dimensional C3T codes computed from SPSA optimization of (25). Similar to previous results on sphere packing [28], the tube packing density decreases rapidly with increase in dimensions.…”

Section: B Optimization Of C3t Encodersupporting

confidence: 66%

“…The fact that constant curvature curves are geodesics on flat tori allows for improved decoder performance since we can project observations onto the torus prior to decoding to the closest point on the curve. That T n/2 is isometric with a n/2hyperrectangle establishes a link to lattice coding methods, but those are not used here; more details on this are available in our previous work [24]. Nearly all of prior studies [17,18] on analog coding with constant curvature curves explicitly make use of the isometry with the hyperrectangle in their decoder construction.…”

Section: A Flat Torimentioning

confidence: 99%

“…where we have used the result from Lemma 1. Therefore, from (24) and utilizing ( 17), (23), and (6), we define our tube-packing objective function for the C3T encoder map as a The optimized value of radii vector r were computed using SPSA for a small sample of even dimensional C3T codes. b The global circumradius (tube radius) ρ G,r as in (26) c The tube density as in ( 24) computed from the optimization of (25) according to (30) and (31).…”

Section: B Optimization Of C3t Encodermentioning

confidence: 99%

“…2) Tube packing. We use the global radius of curvature [24] formulation as our objective function to pack tubes on a flat torus for optimizing the parameters of a constant curvature curve encoder map. We use simultaneous perturbation stochastic approximation as our optimizer because analytic gradients do not exist.…”

Section: Introductionmentioning

confidence: 99%

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Constant Curvature Curve Tube Codes for Low-Latency Analog Error Correction

Taylor¹,

M.²,

Mishra³

et al. 2022

Preprint

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Recent research in ultra-reliable and low latency communications (URLLC) for future wireless systems has spurred interest in short block-length codes. In this context, we introduce a new class of highdimension constant curvature curves codes for analog error correction of independent continuous-alphabet uniform sources. In particular, we employ the circumradius function from knot theory to prescribe insulating tubes about the centerline of constant curvature curves. We then use tube packing density within a hypersphere to optimize the curve parameters. The resulting constant curvature curve tube (C3T) codes possess the smallest possible latency -block-length is unity under bandwidth expansion mapping. Further, the codes provide within 5 dB of Shannon's optimal performance theoretically achievable at the lower range of signal-to-noise ratios and BW expansion factors. We exploit the fact that the C3T encoder locus is a geodesic on a flat torus in even dimensions and a generalized helix in odd dimensions to obtain useful code properties and provide noise-reducing projections at the decoder stage. We validate the performance of these codes using fully connected multi-layer perceptrons that approximate maximum likelihood decoders. For the case of independent and identically distributed uniform sources, we show that analog error correction is advantageous over digital coding in terms of required block-lengths needed to match {signal-to-noise ratio, source-to-distortion ratio} tuples. The best possible digital codes require two to three orders of magnitude higher latency compared to C3T codes, thereby demonstrating the latter's utility for URLLC.

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Section: B Optimization Of C3t Encodersupporting

confidence: 66%

Section: A Flat Torimentioning

confidence: 99%

Section: B Optimization Of C3t Encodermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Constant Curvature Curve Tube Codes for Low-Latency Analog Error Correction

Taylor¹,

M.²,

Mishra³

et al. 2022

Preprint

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“…This method foliates the sphere S 2n−1 by flat manifolds S 1 × • • • × S 1 and distributes points using good packing density lattices in the half-dimension [8], [9]. Other recent contributions to this topic include codes obtained by partitioning the sphere into regions of equal area [10], bounds for constructible codes near the Shannon bound [11], commutative group codes [12], [13] and cyclic group codes [14]. One main challenge for the application of spherical codes is the effective constructiveness for a large range of distances at a reasonable computational cost, which we propose to address in this work.…”

Section: Introductionmentioning

confidence: 99%

Constructive spherical codes in 2^k dimensions

Miyamoto

Earp

Costa

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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We present a new systematic approach to constructing spherical codes in dimensions 2 k , based on Hopf foliations. Using the fact that a sphere S 2n−1 is foliated by manifolds S n−1 cos η × S n−1 sin η , η ∈ [0, π/2], we distribute points in dimension 2 k via a recursive algorithm from a basic construction in R 4. Our procedure outperforms some current constructive methods in several small-distance regimes and constitutes a compromise between achieving a large number of codewords for a minimum given distance and effective constructiveness with low encoding computational cost. Bounds for the asymptotic density are derived and compared with other constructions. The encoding process has storage complexity O(n) and time complexity O(n log n). We also propose a sub-optimal decoding procedure, which does not require storing the codebook and has time complexity O(n log n).

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