arXiv subjects

Eylon E. Krause

Publications and source records attributed to Eylon E. Krause.

3 recordsLinked to original sources

arXiv2026

Shiftfly: Scaling the Accelerator Interconnect Past the Pod with a Shift-Routed Optical Tier

Google's TPU interconnect spent nine generations as a $k$-ary $n$-cube, whose diameter grows as $Θ(N^{1/n})$, before TPU 8i replaced it with Boardfly: a three-tier hierarchy in which every tier is a complete graph, giving pod diameter 7 over 1,152 chips. Boardfly suits a single inference pod but does not extend, because a complete global tier needs $G-1$ optical ports to reach $G$ groups. A 400,000-chip machine would need 12,499 per group against the 40 available, so another hierarchy level must be stacked, and each level costs four chip hops. We propose Shiftfly, which keeps Boardfly's building block and group verbatim and replaces only the global tier with a generalized Kautz digraph, installed as a fixed permutation on the optical circuit switch the fabric already owns. At an identical 40 ports per group, Shiftfly is flat, has guaranteed diameter $\lceil \log_d G \rceil$, routes without tables by a shift register, and addresses content natively. The evaluation is deliberately two-sided. Shiftfly loses at one-pod scale, where Boardfly achieves chip-level diameter 7 against Shiftfly's 11, and wins beyond it, cutting worst-case distance from 23 to 19 hops at 400,000 chips with roughly 2.7x better spectral expansion at equal cost. The redundancy argument that motivated the design does not survive its own evaluation: locality-aware placement supplies most of the achievable saving on shared content, leaving the shift algebra a 2.9% residual, and we report the metric inversion that conceals this. We also measure operability. Replacing a failed group costs the same 40 optical circuits in both designs; deriving the global wiring costs 28 bits of control-plane state against 550,000. Slice allocation is the one regression: an arbitrary induced subset of a shift graph is disconnected, so slices must be instantiated rather than carved.

cs.NI↗

arXiv2026

A Low-PAPR, Synchronization-Robust Non-Coherent Grassmannian Modulation for Optical Communications

Non-coherent Grassmannian (unitary space-time) signaling detects on the received subspace, which is invariant to a branch-side (polarization or mode-coupling) rotation and to a phase that is constant over the coherence block. It therefore needs no carrier-phase or polarization recovery within the block and is robust to phase noise when the per-block phase drift is small, while a multi-branch (polarization or spatial) front end harvests diversity without channel estimation or pilots. However, the Grassmannian-constellation literature usually assumes a distortion-free, linear channel and transmitter and already-acquired symbol timing. This paper closes both gaps while reusing off-the-shelf Grassmannian packings. First, we impose a constant-modulus (low peak-to-average-power-ratio, PAPR) constraint on the constellation and quantify the PAPR/chordal-distance trade-off: a constant-modulus design lowers the 0.1% PAPR from 6.1 dB (unconstrained) to 3.6 dB -- 1.6 dB below 16-QAM (5.2 dB) -- easing the optical modulator linear range and the fiber Kerr-nonlinearity penalty, at a ~1.8 dB cost in high-SNR coding gain. Second, we derive a phase-blind subspace timing-error detector (TED) that exploits the invariance of the GLRT projection energy to the unknown carrier phase, plus a feedforward acquisition metric, supplying clock recovery without prior carrier or polarization recovery. The TED yields a clean S-curve with a stable lock point for roll-offs down to beta=0.1. Under block fading the proposed estimator attains genie-timing SER within a fraction of a dB and recovers full diversity, whereas an uncorrected 0.35-symbol timing offset floors the error rate near 0.4. Results use a symbol-rate block-fading abstraction; full fiber, modulator, and phase-noise modeling is future work. The scheme combines low PAPR with the diversity and phase-recovery-free operation of non-coherent reception.

cs.IT↗

arXiv2026

Geometric Monomial (GEM): a family of rational 2N-differentiable activation functions

The choice of activation function plays a crucial role in the optimization and performance of deep neural networks. While the Rectified Linear Unit (ReLU) remains the dominant choice due to its simplicity and effectiveness, its lack of smoothness may hinder gradient-based optimization in deep architectures. In this work we propose a family of $C^{2N}$-smooth activation functions whose gate follows a log-logistic CDF, achieving ReLU-like performance with purely rational arithmetic. We introduce three variants: GEM (the base family), E-GEM (an $ε$-parameterized generalization enabling arbitrary $L^p$-approximation of ReLU), and SE-GEM (a piecewise variant eliminating dead neurons with $C^{2N}$ junction smoothness). An $N$-ablation study establishes $N=1$ as optimal for standard-depth networks, reducing the GELU deficit on CIFAR-100 + ResNet-56 from 6.10% to 2.12%. The smoothness parameter $N$ further reveals a CNN-transformer tradeoff: $N=1$ is preferred for deep CNNs, while $N=2$ is preferred for transformers. On MNIST, E-GEM ties the best baseline (99.23%). On CIFAR-10 + ResNet-56, SE-GEM ($ε=10^{-4}$) surpasses GELU (92.51% vs 92.44%) -- the first GEM-family activation to outperform GELU. On CIFAR-100 + ResNet-56, E-GEM reduces the GELU deficit from 6.10% (GEM $N=2$) to just 0.62%. On GPT-2 (124M), GEM achieves the lowest perplexity (72.57 vs 73.76 for GELU), with GEM $N=1$ also beating GELU (73.32). On BERT-small, E-GEM ($ε=10$) achieves the best validation loss (6.656) across all activations. The $ε$-parameterization reveals a scale-dependent optimum: small $ε$ ($10^{-4}$--$10^{-6}$) for deep CNNs and larger transformers, with the special case of small transformers (BERT-small) benefiting from large $ε$ ($ε=10$) due to its limited depth and unconstrained gradients.

cs.LG↗