Vladimer Khasia

Vladimer Khasia

Coordinate-based neural networks have emerged as a powerful tool for representing continuous physical fields, yet they face two fundamental pathologies: spectral bias, which hinders the learning of high-frequency dynamics, and the curse of dimensionality, which causes parameter explosion in discrete feature grids. We propose the Adaptive Vekua Cascade (AVC), a hybrid architecture that bridges deep learning and classical approximation theory. AVC decouples manifold learning from function approximation by using a deep network to learn a diffeomorphic warping of the physical domain, projecting complex spatiotemporal dynamics onto a latent manifold where the solution is represented by a basis of generalized analytic functions. Crucially, we replace the standard gradient-descent output layer with a differentiable linear solver, allowing the network to optimally resolve spectral coefficients in a closed form during the forward pass. We evaluate AVC on a suite of five rigorous physics benchmarks, including high-frequency Helmholtz wave propagation, sparse medical reconstruction, and unsteady 3D Navier-Stokes turbulence. Our results demonstrate that AVC achieves state-of-the-art accuracy while reducing parameter counts by orders of magnitude (e.g., 840 parameters vs. 4.2 million for 3D grids) and converging 2-3x faster than implicit neural representations. This work establishes a new paradigm for memory-efficient, spectrally accurate scientific machine learning. The code is available at https://github.com/VladimerKhasia/vecua.

Vladimer Khasia

Sequence modeling universally relies on discrete subword tokenization to circumvent the $\mathcal{O}(N^2)$ computational intractability of native byte-level attention. However, this heuristic quantization imposes artificial morphological boundaries, enforces vocabulary dependence, and fractures the continuity of the optimization landscape. To resolve this dichotomy, we introduce \textbf{HoloByte}: a strictly tokenizer-free framework utilizing Continuous Hyperspherical Distillation. HoloByte partitions discrete byte sequences into fixed-capacity chunks and projects them into a continuous, strictly bounded hyperspherical manifold via an invertible, dimension-preserving orthogonal rotation operator. This spatial superposition allows a macroscopic transformer to operate exclusively on compressed continuous representations, formally reducing the exact attention time complexity from $\mathcal{O}(N^2D)$ to $\mathcal{O}\left( \frac{N^2}{W^2}D + ND^2 \right)$. A localized causal micro-decoder subsequently unbinds these representations to compute exact byte-level distributions. To govern this continuous trajectory, we propose a dual-objective formulation incorporating a mathematically precise Holographic Latent Mean Squared Error, which strictly bounds the gradient and guarantees asymptotic stability. Theoretically, we derive the minimal embedding dimension $D = Ω(W \ln |\mathcal{V}|)$ required to ensure error-free discrete recovery from the continuous manifold. Empirically, under strictly matched parameter constraints, HoloByte is systematically outperforming a comparable discrete Byte-Pair Encoding (BPE) baseline. These results establish Continuous Hyperspherical Distillation as a mathematically rigorous and computationally tractable foundation for vocabulary-invariant sequence modeling. The code is available at https://github.com/VladimerKhasia/HoloByte

Vladimer Khasia

The activation memory required for exact backpropagation scales linearly with network depth, context length, and feature dimensionality, forming an O(L * BN ) spatial bottleneck (where B is the sequence-batch cardinality and N is the feature dimension). This constraint historically throttles the scaling of deep neural networks. While randomized automatic differentiation attempts to mitigate this, it historically suffers from catastrophic variance. In this paper, we introduce BASIS (Balanced Activation Sketching with Invariant Scalars), an efficient backpropagation algorithm that fully decouples activation memory from the batch and sequence dimensions. BASIS propagates the exact error signal (dX) to preserve flawless gradient flow, but computes the weight updates (dW) using massively compressed rank-R tensors. To solve the foundational instability of sketched gradients, we propose two novel mechanisms: Balanced Hashing, which strictly eliminates off-diagonal collision variance, and Invariant Scalars, a principled bias-variance tradeoff that deterministically preserves the exact continuous energy norm of the spatial geometry. Theoretically, BASIS reduces activation memory to O(L * RN ) and heavily decreases the backward pass matrix-multiplication footprint. Empirically, training a GPT architecture for 50,000 steps validates our theoretical guarantees: at R = 32, BASIS achieves parity with (and marginally outperforms) exact backpropagation validation loss (6.575 vs. 6.616), acting as an implicit regularizer. Remarkably, the stabilized magnitude trajectory allows the model to converge smoothly even under extreme spatial compression (R = 1), proving the extreme robustness of the estimator. The code is available at https://github.com/VladimerKhasia/basis

Vladimer Khasia

Standard Transformer architectures rely heavily on dense linear transformations, treating feature projection as a monolithic, full-rank operation. We argue that this formulation is inefficient and lacks the structural inductive bias necessary for distinguishing between local feature preservation and global context integration. To address this, we introduce the Hybrid Dual-Path Linear (HDPL) operator, which decomposes the affine transformation into two topologically distinct pathways: a sparse block-diagonal component for high-rank local processing, and a low-rank Variational Autoencoder (VAE) bottleneck for global context regularization. By "surgically" replacing specific projections (Query, Key, Value, Gate, Up) with HDPL operators while retaining standard dense layers for aggregation (Output, Down), we achieve a superior balance of efficiency and representational power. Experiments on the FineWeb-Edu dataset demonstrate that the HDPL architecture outperforms a standard Llama-style baseline, reducing validation loss while simultaneously reducing parameter count by 6.8%. Beyond immediate performance gains, we discuss how the explicit materialization of a probabilistic latent space within the Transformer backbone serves as a vital architectural affordance, offering new pathways for inference-time or hypernetwork induced control, continual adaptation, interpretability, and cross-model or cross-modal synchronization. The code is available at https://github.com/VladimerKhasia/HDPL

Vladimer Khasia

Mixture of Experts (MoE) models scale capacity but often suffer from representation collapse and gradient instability. We propose Dynamic Subspace Composition (DSC), a framework that approximates context-dependent weights via a state-dependent, sparse expansion of a shared basis bank. Formally, DSC models the weight update as a residual trajectory within a Star- Shaped Domain, employing a Magnitude-Gated Simplex Interpolation to ensure continuity at the identity. Unlike standard Mixture-of-LoRAs, which incurs O(M rd) parameter complexity by retrieving independent rank-r matrices, DSC constructs a compositional rank-K approximation from decoupled unit-norm basis vectors. This reduces parameter complexity to O(M d) and memory traffic to O(Kd), while Frame-Theoretic regularization and spectral constraints provide rigorous worst-case bounds on the dynamic update. The code is available at https://github. com/VladimerKhasia/DSC

Vladimer Khasia

Scaling sequence modeling to extreme contexts requires balancing computational efficiency with representational expressivity. While Transformers provide precise retrieval via the attention mechanism, their quadratic $\mathcal{O}(T^2)$ complexity limits their application to long-horizon tasks. In this work, we propose the \textbf{Spectral-Window Hybrid (SWH)}, an architecture that decouples sequence modeling into two \textit{parallel} streams: a global branch utilizing the Convolution Theorem to model long-range decay dynamics in $\mathcal{O}(T \log T)$ time, and a local branch employing sliding-window attention for token interactions within a bounded context. By aggregating these representations, SWH avoids the computational bottleneck of global attention while retaining local precision. We demonstrate that SWH matches the perplexity of standard Transformers on short contexts while enabling efficient linear scaling to extended sequences. The code is available at https://github.com/VladimerKhasia/SWH

Vladimer Khasia

We present DeepVekua, a hybrid architecture that unifies geometric deep learning with spectral analysis to solve partial differential equations (PDEs) in sparse data regimes. By learning a diffeomorphic coordinate transformation that maps complex geometries to a latent harmonic space, our method outperforms state-of-the-art implicit representations on advection-diffusion systems. Unlike standard coordinate-based networks which struggle with spectral bias, DeepVekua separates the learning of geometry from the learning of physics, solving for optimal spectral weights in closed form. We demonstrate a 100x improvement over spectral baselines. The code is available at https://github.com/VladimerKhasia/vekuanet.

Vladimer Khasia

We present Primal, a deterministic feature mapping framework that harnesses the number-theoretic independence of prime square roots to construct robust, tunable vector representations. Diverging from standard stochastic projections (e.g., Random Fourier Features), our method exploits the Besicovitch property to create irrational frequency modulations that guarantee infinite non-repeating phase trajectories. We formalize two distinct algorithmic variants: (1) StaticPrime, a sequence generation method that produces temporal position encodings empirically approaching the theoretical Welch bound for quasi-orthogonality; and (2) DynamicPrime, a tunable projection layer for input-dependent feature mapping. A central novelty of the dynamic framework is its ability to unify two disparate mathematical utility classes through a single scaling parameter σ. In the low-frequency regime, the method acts as an isometric kernel map, effectively linearizing non-convex geometries (e.g., spirals) to enable high-fidelity signal reconstruction and compressive sensing. Conversely, the high-frequency regime induces chaotic phase wrapping, transforming the projection into a maximum-entropy one-way hash suitable for Hyperdimensional Computing and privacy-preserving Split Learning. Empirical evaluations demonstrate that our framework yields superior orthogonality retention and distribution tightness compared to normalized Gaussian baselines, establishing it as a computationally efficient, mathematically rigorous alternative to random matrix projections. The code is available at https://github.com/VladimerKhasia/primal

Vladimer Khasia

The pursuit of world model based artificial intelligence has predominantly relied on projecting high-dimensional observations into parameterized latent spaces, wherein transition dynamics are subsequently learned. However, this conventional paradigm is mathematically flawed: it merely displaces the manifold learning problem into the latent space. When the underlying data distribution shifts, the latent manifold shifts accordingly, forcing the predictive operator to implicitly relearn the new topological structure. Furthermore, by classical approximation theory, positive operators like dot product attention inevitably suffer from the saturation phenomenon, permanently bottlenecking their predictive capacity and leaving them vulnerable to the curse of dimensionality. In this paper, we formulate a mathematically rigorous paradigm for world model construction by redefining the core predictive mechanism. Inspired by Ryan O'Dowd's foundational work we introduce Spherical Kernel Operator (SKO), a framework that replaces standard attention. By projecting the unknown data manifold onto a unified ambient hypersphere and utilizing a localized sequence of ultraspherical (Gegenbauer) polynomials, SKO performs direct integral reconstruction of the target function. Because this localized spherical polynomial kernel is not strictly positive, it bypasses the saturation phenomenon, yielding approximation error bounds that depend strictly on the intrinsic manifold dimension q, rather than the ambient dimension. Furthermore, by formalizing its unnormalized output as an authentic measure support estimator, SKO mathematically decouples the true environmental transition dynamics from the biased observation frequency of the agent. Empirical evaluations confirm that SKO significantly accelerates convergence and outperforms standard attention baselines in autoregressive language modeling.

Vladimer Khasia

Implicit Neural Representations (INRs) have emerged as a powerful paradigm for parameterizing physical fields, yet they often suffer from spectral bias and the computational expense of non-convex optimization. We introduce the Vekua Layer (VL), a differentiable spectral method grounded in the classical theory of Generalized Analytic Functions. By restricting the hypothesis space to the kernel of the governing differential operator -- specifically utilizing Harmonic and Fourier-Bessel bases -- the VL transforms the learning task from iterative gradient descent to a strictly convex least-squares problem solved via linear projection. We evaluate the VL against Sinusoidal Representation Networks (SIRENs) on homogeneous elliptic Partial Differential Equations (PDEs). Our results demonstrate that the VL achieves machine precision ($\text{MSE} \approx 10^{-33}$) on exact reconstruction tasks and exhibits superior stability in the presence of incoherent sensor noise ($\text{MSE} \approx 0.03$), effectively acting as a physics-informed spectral filter. Furthermore, we show that the VL enables "holographic" extrapolation of global fields from partial boundary data via analytic continuation, a capability absent in standard coordinate-based approximations.