arXiv subjects

Mahesh Godavarti

Publications and source records attributed to Mahesh Godavarti.

13 recordsLinked to original sources

arXiv2026

Labeled Incidence Structures for Native Transformer Modeling of Text, Knowledge Graphs, and Hypergraphs

Current Transformer interfaces index tokens by one or more integer coordinates, which determine their addresses inside attention. In RoPE and its multi-axis or hierarchical variants, the resulting address has the form $A(i)=R_1^{i_1}R_2^{i_2}R_3^{i_3}$, where the exponents are integer coordinates assigned after choosing a serialized token layout. When Transformers process new or large collections of data, this addressing scheme can produce unseen offsets or coordinate combinations, push repositories toward retrieve-and-serialize pipelines, and force new entities, records, or repository items to be represented by long token strings or identifier embeddings not seen in training. We introduce labeled incidence structures (LIS), in which each participating token or value is an endpoint with content $x$ and a structural index $i$. The index can include local position, relation role, relation instance, text unit, field, or content-derived identity. The model maps this index to a structural address $A(i)$, so adding new tokens, facts, text units, or repository items applies the same learned address rule to structural and content coordinates rather than requiring larger integer coordinates, unseen coordinate combinations, or new identifier embeddings. Attention scores endpoints $i,j$ using $q_i^\top P_{j\to i}k_j$, where journey consistency forces $P_{j\to i}=A(i)^{-1}A(j)$. When $i$ has several coordinates, such as position, role, and instance, coordinate independence is equivalent to factoring $A(i)$ into one address factor per coordinate. This recovers RoPE, RoPE-2D, and HiRoPE as special cases. This allows knowledge-graph (KG) roles, fact instances, and text units to enter the attention score directly. In controlled shallow diagnostics, the LIS address interface is implemented inside ordinary Transformer attention and yields promising results across text, KG, and $n$-ary settings.

cs.LG↗

arXiv2026

Content-Based Addressing for Long Context

Rotary position embedding (RoPE) uses each token's integer position to determine the rotation applied inside attention. This works well for local token order, but increasing context length creates a positional train-test mismatch: RoPE produces relative rotations at offsets not seen during training. Methods that rescale, interpolate, randomize, or bias positions specify how attention handles those offsets, but still derive positional information from a growing token counter. We instead divide a token stream into units, retain ordinary RoPE positions within each unit, and assign every completed unit an address computed from its content. Adding units then applies the same learned map to new content rather than extending a positional range or an identifier table. We prove that this construction preserves local RoPE exactly, leaves the attention comparison between two fixed tokens unchanged when other units are inserted or reordered, and does not create new relative rotations merely because more units are added. In a character-level Tiny Shakespeare diagnostic, all-token validation perplexity remains approximately constant from contexts of 256 to 4096 characters. A second diagnostic shows that content-based addressing can retrieve and use information from multiple serialized facts. These are controlled shallow experiments, not scale benchmarks, but they support a direct prescription: use position to address locally and content to address across units.

cs.CL↗

arXiv2026

Journey Operators for Structured Multi-Axis Composition

Many kinds of data have structure along one or more axes: words in a sentence, pixels in an image, nodes in a tree, frames in audio, or cells in a 3D volume. Along one axis, order matters: "the dog bit the man" is different from "the man bit the dog." Across independent axes, however, neither composition nor movement should depend on the order of axes: in an image, composing right then down should give the same result as composing down then right, and moving right then down should describe the same relative position as moving down then right. We develop a framework for modeling this kind of multi-axis structure. Each data item carries its content together with a small transformation for each axis. A path connecting two positions defines a journey; the journey operator is the product of per-axis transformations along that path, governing both how data composes along the path and how relative position is described. When the transformations are fixed, our framework recovers Rotary Position Embedding (RoPE) and its multi-dimensional variants. When they depend on the data, the model gains a content-adaptive positional inductive bias. We show exactly when these paths are well-defined: both composition and movement across axes are path-independent precisely when the axis transformations commute. We also prove that, under the stated toral-frame symmetry, cocycle, bilinearity, and norm-preservation assumptions, the resulting pairwise scoring rule must take the form of block-wise rotations, explaining why RoPE-like methods arise naturally. Finally, we use this theory to design JoFormer, a model for value aggregation, and relate it to attention and state-space models (SSMs). Initial experiments across vision, language, and length generalization suggest that these inductive biases can have observable consequences in practice.

cs.LG↗

arXiv2026

Cumsum-Composable Phase Transport for Low-Cost Streaming Keyword Spotting

State-space sequence models are attractive for streaming speech because they maintain compact recurrent state, but scan-style training kernels can have unfavorable constants for short audio tasks. We study cumsum-composable phase transport, a streaming-native temporal layer for keyword spotting. Each layer projects acoustic frames to complex channels, transports them by learned unitary rotations, accumulates a finite window using prefix differences, and applies a gated residual update. The same prefix representation gives exact batched training with ordinary cumulative sums and exact online inference with one prefix update per frame. Unitary transport is the key constraint: inverse rotations have norm one, keeping prefix terms well conditioned while memory is supplied by windows or block readouts. On Google Speech Commands v2 with 12 labels, mel+cumsum models retain competitive accuracy with compact baselines. The strongest single-seed run reaches 97.3\% test accuracy; a 51.6K-parameter tied model also reaches 97.3\%, and a 24.8K tied model reaches 96.8\% versus 97.1\% for a 25.6K MelCNNMaxPool baseline. In a matched cumsum-versus-scan benchmark, cumsum+window gives comparable accuracy, 94.82\% versus 94.33\%, while training 1.07x faster and reducing single-example latency from 7.09 ms to 5.01 ms on a Tesla T4. These results support cumsum phase transport as a simple low-cost temporal primitive for streaming keyword spotting.

cs.SD↗

arXiv2026

The Context-Ready Transformer

We introduce the context-ready transformer, a new recurrent neural network architecture built from a D-layer transformer block that pre-contextualizes each token before it enters the block. During left-to-right generation, a correction network combines the previous position's block output -- a cached summary of past context -- with the current token embedding, so the tokenenters the block already contextualized rather than as a raw embedding. At sequential inference, the correction chain makes the architecture a recurrent neural network. For training, we unroll the correction process K times over the full sequence, processing all positions in parallel at each step. A pretrained transformer can also be converted to a context-ready model by adding a zero-initialized correction FFN and fine-tuning. We evaluate across widths, depths, block sizes, and two datasets, with all comparisons against standard transformers, variants, and ablations. A D=5 model beats a 12-layer transformer while generating 1.7x faster on an A100. With K=10, a single-layermodel (D=1) beats a 6-layer transformer with a 2.6x inference speedup, and sequential inference matches parallel K=10 to within 0.01 PPL. The architecture benefits most from wide representations and long contexts. On a pointer-chasing task, D=1 trained with BPTT solves all 10 composition levels, while standard transformers exhibit staircase-like depth dependence.

cs.CL↗

arXiv2026

Diversity vs Degrees of Freedom in Gaussian Fading Channels

The classical definitions extract degrees of freedom (DOF) via $C(\snr)/\log\snr$ and diversity (DIV) via $-\log P_e(\snr)/\log\snr$, using $\log\snr$ as the common gauge for both. These ratios hide a two-step process: first, identify the gauge on which capacity or reliability actually grows; second, normalize the coefficient on that gauge by the appropriate atom. For coherent multiple-input multiple-output (MIMO) both gauges happen to be $\log\snr$ and both atom coefficients happen to be one. This paper shows that the two-step process is necessary outside this calibration case and makes it explicit using a Bhattacharyya-frontier construction. A capacity--packing sandwich theorem shows that fixed-resolution output-law packing and covering recover the capacity gauge, and a binary-endpoint theorem shows that the two-message Bhattacharyya frontier identifies the zero-rate diversity gauge. Endpoint DOF and endpoint DIV are obtained by dividing the raw coefficient on the identified gauge by the corresponding atom coefficient. For fixed deterministic channel matrix~$H$, the capacity gauge is $\log\snr$ with endpoint DOF $T\,\mathrm{rank}(H)$, while the zero-rate diversity gauge is $\snr$ with endpoint DIV $Tσ_1^2(H)$, making fixed-$H$ a cross-gauge channel. For noncoherent scalar fast fading with $N$~receive antennas, the capacity gauge is $\log\log\snr$ with DOF~$1$, while the zero-rate diversity gauge is $\log\snr$ with endpoint DIV~$N$; the exact load-$r$ frontier gauge is $(\log\snr)^{1-r}$. The framework recovers the same-gauge cases, coherent Rayleigh MIMO and noncoherent block fading, with zero-rate DIV~$MN$. Audit tables separate exact results from lower bounds and open problems.

cs.IT↗

arXiv2026

Why Do Accumulated Transformations Extrapolate?

PaTH Attention showed that replacing RoPE's position-indexed rotations with accumulated data-dependent Householder reflections yields strong length extrapolation, though performance degrades at extreme context lengths. We ask whether this depends on Householder-specific structure or reflects a general property of accumulated transformations along source-to-query paths. We study a simpler variant keeping RoPE's block-diagonal SO(2) rotations but replacing position-indexed angles with accumulated token-dependent ones. It shows the same pattern: improved extrapolation then degradation at long contexts. We prove the result extends to accumulated orthogonal transformations satisfying certain regularity conditions: their products become incoherent after finitely many steps, suppressing attention to distant tokens. Accumulated rotations of queries and keys create a finite mixing window independent of context length; per-token suppression learned in training transfers unchanged to any evaluation length, and high-dimensional concentration produces a score gap suppressing far tokens while near-route transport preserves the target signal. Conversely, a lower bound shows accumulated rotations must eventually degrade: as the far set grows, no rotations preserve the near signal without explicit far-mass control. For SO(2) rotations, rotating values too makes residual far contributions combine incoherently, extending the range. Controlled experiments support these predictions: random accumulated rotations substantially improve extrapolation over RoPE, learned token-dependent rotations maintain near-training-length perplexity far beyond the training context, and rotating values helps over queries and keys alone. Rotation-only models still degrade at extreme lengths, while ALiBi stays length-stable, consistent with the need for far-mass control.

cs.LG↗

arXiv2026

Convergence of Differential Entropies -- II

We show that under convergence in measure of probability density functions, differential entropy converges whenever the entropy integrands $f_n |\log f_n|$ are uniformly integrable and tight -- a direct consequence of Vitali's convergence theorem. We give an entropy-weighted Orlicz condition: $\sup_n \int f_n\, Ψ(|\log f_n|) < \infty$ for a single superlinear~$Ψ$, strictly weaker than the fixed-$α$ condition of Godavarti and Hero (2004). We also disprove the Godavarti-Hero conjecture that $α> 1$ could be replaced by $α_n \downarrow 1$. We recover the sufficient conditions of Godavarti--Hero, Piera--Parada, and Ghourchian-Gohari-Amini as corollaries. On bounded domains, we prove that uniform integrability of the entropy integrands is both necessary and sufficient -- a complete characterization of entropy convergence.

cs.IT↗

arXiv2026

Knowledge Graph and Hypergraph Transformers with Repository-Attention and Journey-Based Role Transport

We present a concise architecture for joint training on sentences and structured data while keeping knowledge and language representations separable. The model treats knowledge graphs and hypergraphs as structured instances with role slots and encodes them into a key-value repository that a language transformer can attend over. Attention is conditioned by journey-based role transport, which unifies edge-labeled KG traversal, hyperedge traversal, and sentence structure. We outline a dual-stream architecture, hierarchical layer groups with instance-local, neighborhood, and global mixing attention, retrieval over a separate repository, and multi-task objectives spanning masked language modeling, link prediction, and role-consistency denoising. The result is an explicit, inspectable separation between linguistic context and structured knowledge, while still enabling tight alignment through cross-attention.

cs.LG↗

arXiv2025

JoFormer (Journey-based Transformer): Theory and Empirical Analysis on the Tiny Shakespeare Dataset

Transformers have demonstrated remarkable success in sequence modeling, yet effectively incorporating positional information remains a challenging and active area of research. In this paper, we introduce JoFormer, a journey-based Transformer architecture grounded in a recently proposed non-commutative algebra for composing transformations across positions. JoFormer represents relative positions through learnable directional transforms that are sequentially composed along the input, thereby extending and generalizing existing approaches based on relative position representations. We derive the JoFormer attention mechanism from first principles and show that it subsumes standard methods such as rotary transformations as special cases. To evaluate its effectiveness, we compare JoFormer to the RoFormer baseline on the Tiny Shakespeare character-level language modeling task. Our results demonstrate that JoFormer consistently achieves lower perplexity and faster convergence, highlighting the advantages of its more expressive, journey-based treatment of position. Notably, the per-token JoFormer is still a primitive, conceptual variant with layer-independent angles, yet it already demonstrates strong performance-underscoring its promise as a proof of concept for more expressive architectures. We conclude by discussing how JoFormer offers a principled approach to integrating positional structure into Transformer architectures. The code used in this work is available at https://github.com/mahesh-godavarti/joformer.

cs.LG↗

arXiv2025

Directional Non-Commutative Monoidal Embeddings for MNIST

We present an empirical validation of the directional non-commutative monoidal embedding framework recently introduced in prior work~\cite{Godavarti2025monoidal}. This framework defines learnable compositional embeddings using distinct non-commutative operators per dimension (axis) that satisfy an interchange law, generalizing classical one-dimensional transforms. Our primary goal is to verify that this framework can effectively model real data by applying it to a controlled, well-understood task: image classification on the MNIST dataset~\cite{lecun1998gradient}. A central hypothesis for why the proposed monoidal embedding works well is that it generalizes the Discrete Fourier Transform (DFT)~\cite{oppenheim1999discrete} by learning task-specific frequency components instead of using fixed basis frequencies. We test this hypothesis by comparing learned monoidal embeddings against fixed DFT-based embeddings on MNIST. The results show that as the embedding dimensionality decreases (e.g., from 32 to 8 to 2), the performance gap between the learned monoidal embeddings and fixed DFT-based embeddings on MNIST grows increasingly large. This comparison is used as an analytic tool to explain why the framework performs well: the learnable embeddings can capture the most discriminative spectral components for the task. Overall, our experiments confirm that directional non-commutative monoidal embeddings are highly effective for representing image data, offering a compact learned representation that retains high task performance. The code used in this work is available at https://github.com/mahesh-godavarti/directional_composition_mnist.

cs.LG↗

arXiv2025

Directional Non-Commutative Monoidal Structures with Interchange Law via Commutative Generators

We introduce a novel framework consisting of a class of algebraic structures that generalize one-dimensional monoidal systems into higher dimensions by defining per-axis composition operators subject to non-commutativity and a global interchange law. These structures, defined recursively from a base case of vector-matrix pairs, model directional composition in multiple dimensions while preserving structural coherence through commutative linear operators. We show that the framework that unifies several well-known linear transforms in signal processing and data analysis. In this framework, data indices are embedded into a composite structure that decomposes into simpler components. We show that classic transforms such as the Discrete Fourier Transform (DFT), the Walsh transform, and the Hadamard transform are special cases of our algebraic structure. The framework provides a systematic way to derive these transforms by appropriately choosing vector and matrix pairs. By subsuming classical transforms within a common structure, the framework also enables the development of learnable transformations tailored to specific data modalities and tasks.

cs.LG↗

arXiv2025

Directional Non-Commutative Monoidal Structures for Compositional Embeddings in Machine Learning

We introduce a new algebraic structure for multi-dimensional compositional embeddings, built on directional non-commutative monoidal operators. The core contribution of this work is this novel framework, which exhibits appealing theoretical properties (associativity along each dimension and an interchange law ensuring global consistency) while remaining compatible with modern machine learning architectures. Our construction defines a distinct composition operator circ_i for each axis i, ensuring associative combination along each axis without imposing global commutativity. Importantly, all axis-specific operators commute with one another, enforcing a global interchange law that enables consistent crossaxis compositions. This is, to our knowledge, the first approach that provides a common foundation that generalizes classical sequence-modeling paradigms (e.g., structured state-space models (SSMs) and transformer self-attention) to a unified multi-dimensional framework. For example, specific one-dimensional instances of our framework can recover the familiar affine transformation algebra, vanilla self-attention, and the SSM-style recurrence. The higher-dimensional generalizations naturally support recursive, structure-aware operations in embedding spaces. We outline several potential applications unlocked by this structure-including structured positional encodings in Transformers, directional image embeddings, and symbolic modeling of sequences or grids-indicating that it could inform future deep learning model designs. We formally establish the algebraic properties of our framework and discuss efficient implementations. Finally, as our focus is theoretical, we include no experiments here and defer empirical validation to future work, which we plan to undertake.

cs.LG↗