It is demonstrated that finite-difference time-domain dynamics can be reproduced with field variables restricted to the ternary alphabet, and its extension to acoustics, Virieux-type elastodynamics and Schrodinger-equation solvers points to a broader class of quantised physics solvers for resource-constrained and specialised hardware.
Abstract
Motivated by the rapid development of quantised large language models, which substantially reduce computational cost and enable efficient artificial intelligence on resource-constrained and consumer hardware, I demonstrate that finite-difference time-domain (FDTD) dynamics can be reproduced with field variables restricted to the ternary alphabet ${-1,0,+1}$. The resulting update requires no run-time multiplication or floating-point arithmetic and represents each field component with just $\log_2 3\simeq1.58$ bits. A true state accumulator performs the integration, while a second-order noise-shaped encoder with an independent error register performs the quantisation. The Courant number S serves simultaneously as an exact fixed-point ratio and as the encoder's oversampling ratio. Ternary FDTD converges to the standard Yee scheme in the small-S limit while substantially reducing state storage and arithmetic complexity. Its extension to acoustics, Virieux-type elastodynamics and Schrodinger-equation solvers points to a broader class of quantised physics solvers for resource-constrained and specialised hardware.
Noise-shaped one-bit coefficients in normalized discrete polynomial Fourier extension are studied, and exact orthogonality identities, fourth-moment formulas, local kernel estimates, and oscillatory transfer bounds are established.
When can additional low-bit residual computation replace missing numerical precision for a fixed input-output map? We model a quantized residual system over a fixed horizon as a pure schedule selecting fields from a declared low-bit operation library, and use relaxed controls to characterize its infinite-depth limit. The distance from the target to the closed relaxed reachable set is the exact structural floor: no increase in depth can remove it for that library. Pure schedules approach the relaxed class at rate $O(D^{-1})$ under bounded-variation time dependence and $O(D^{-\vartheta}+D^{-1})$ under Holder dependence of exponent $\vartheta$. Execution arithmetic can reverse this conclusion: full-state write-back introduces a $D\rho_z$ penalty and can freeze residual updates, whereas increment error feedback replaces this growth by a bounded carry term and obeys an exact common-lattice conservation law. A fixed-teacher converse makes this rate sharp: for coherent depth-$L$ first-order high-precision comparators, accuracy matching requires $D=\Theta(L)$. Learned codebooks add a metadata resource, while state-dependent routing introduces hybrid event conditions. Verified primal and dual bounds yield feasible, impossible, or unresolved decisions before training. Companion software implements the workflow, and Lean 4 machine-checks the exact discrete core. Depth replaces precision only relative to a declared library, horizon, execution semantics, and routing model.
Processing in memory (PIM) offers a compelling pathway to overcome the data movement bottleneck in modern AI and data-centric systems. This work introduces MITRA, a reconfigurable magnetic tunnel junction (MTJ)-based in-memory architecture that leverages stochastic computing (SC) to implement a broad class of transcendental and nonlinear functions directly within memory. By combining stochastic bit-stream processing with compact finite-state-machines (FSMs) embedded in MTJ-FinFET logic-in-memory structures, the proposed design achieves low-latency and power-efficient computation without external datapaths, unlike the binary counterparts. Circuit-level simulations in 14-nm FinFET technology verify correct state transitions, stable stochastic outputs, and predictable power profiles. Extensive evaluations demonstrate high accuracy even with short bit-streams. We further integrate the design into a neural-network classifier and develop an FSM-aware training strategy that compensates for approximation errors, achieving up to 96.9% classification accuracy on the UCI Optical Digit benchmark. Overall, MITRA provides a compact, reconfigurable platform for nonlinear processing in next-generation edge AI systems.
Farzad Razi, M. Moghadam, M. Najafi et al.· Proceedings of the ACM/IEEE...· 0 citations
Approximate message passing (AMP)-type algorithms are widely used for signal recovery in high-dimensional noisy linear systems. Recently, a framework called memory AMP (MAMP) was introduced, offering a new approach to incorporating memory terms within AMP algorithms. Building on this, a low-complexity gradient descent MAMP (GD-MAMP) was proposed for right-unitarily invariant matrices. In this paper, we first address an overflow problem in GD-MAMP caused by intermediate variables exceeding the floating-point range, which typically occurs when the condition number is large. Second, we propose two low-complexity variants of GD-MAMP: one replaces full-length memory with partial memory, while the other reduces the number of matrix-vector products per iteration by $1/3$ (from three to two). Neither degrades the convergence speed notably. Third, we develop a general gradient-based formulation for designing MAMP algorithms. This formulation recovers warm-started conjugate gradient VAMP (WS-CG-VAMP) as a special case. Furthermore, we show that the computation of the orthogonalization parameters in this formulation can suffer from catastrophic cancellation, which explains the finite-precision instability of WS-CG-VAMP. Finally, we derive an equivalent reformulation, termed WS-CG-VAMP(r), which reduces the number of matrix-vector products by up to $50\%$. Measured by matrix-vector products, GD-MAMP converges faster for small condition numbers, whereas WS-CG-VAMP(r) converges faster for large ones under high-precision arithmetic but may diverge in IEEE double precision due to catastrophic cancellation.
Most competitive 4-bit LLM research pipelines now open the same way: apply a linear, function-preserving transform (rotation, scaling, permutation, non-orthogonal affine) so the outlier mass sits more favorably against the group scales, and only then round. Yet we are aware of no survey dedicated to this transform stage, and its literature is quietly re-deriving an older theory. We identify and formalize the principle that organizes it, the Great Inversion: allocation-flexible coding rewards energy concentration, whereas the grouped shared-scale quantization a deployed matrix instruction performs rewards within-group flattening. Classical transform coding (1963: decorrelate, allocate bits, quantize) spends different bits per coordinate at a fixed total rate; for a Gaussian source at high rate the Karhunen-Loeve transform's concentration minimizes distortion. A deployed operand tile instead carries one absolute-maximum scale per group and equal bits everywhere, with no allocation; on a uniform grid that objective rewards flattening, approached by Hadamard incoherence. We prove that opposition under within-group majorization: the prescriptions point in opposite directions, each backed by a proof against its own objective, and for a generic spectrum no optimality guarantee transfers. A second axis is the number format: the non-uniform FP4 grid makes flattening buy less, MXFP4's power-of-two block scale still rewards a rotation confined to that block, and NVFP4's mantissa-carrying scale largely removes that pull, so the target pole depends jointly on allocation regime and format. We survey 200 works to a June 2026 cutoff; classify 43 transform methods by structure, data-awareness, searched-versus-constructed, and runtime cost; record, where reported, how they compose with GPTQ rounding; distill a first-choice guide by deployment regime; and close with the open problems it exposes.
Lossy audio compression algorithms traditionally rely on psychoacoustic modeling and frequency-domain representations (e.g., MP3, AAC, and Opus) to discard information that is imperceptible to the human auditory system. While highly effective, these approaches are computationally complex and domain-specific. In this paper, we present the design and mathematical formulation of AudioTQ, a data-oblivious lossy audio codec that operates directly in the time domain. Inspired by Large Language Model (LLM) weight quantization techniques (specifically the TurboQuant framework), AudioTQ uniformizes volatile time-domain amplitudes into a predictable standard normal distribution using an orthonormal, randomized Fast Walsh-Hadamard Transform (FWHT) rotation. This enables coordinate-wise scalar quantization using an offline-trained, MSE-optimal 6-bit Lloyd-Max quantizer, augmented by a 1-bit Quantized Joint Least-Squares (QJL) residual correction layer. The resulting 7-bit virtual indices are packed into native 8-bit containers, aligning with standard CPU register boundaries to ensure real-time single-threaded execution without hardware parallel accelerators. We detail the bitwise reconstruction of 24-bit studio stems, analyze the butterfly network of the FWHT, derive the mathematical failure modes under sparse inputs, and present benchmarks showing up to 74.4% physical size reduction alongside a Signal-to-Quantization-Noise Ratio (SQNR) of ~30 dB.