Ratio

Zero-shot text-to-speech synthesizes new utterances in a speaker's voice from a short reference recording. Voice cloning requires accurate content and preserved speaker identity, but supervised acoustic-token prediction does not directly optimize these waveform-level properties. Reward-based post-training addresses this mismatch, but in discrete diffusion, token choices and reveal positions jointly define the sampling trajectory, complicating alignment. We introduce RAWD-TTS (Ratio-free Advantage-Weighted Denoising), which scores decoded samples with recognition and speaker rewards and uses group-relative advantages to weight masked-token reconstruction of those samples, without reverse-trajectory likelihoods or target audio. On 500 Russian CV3-Eval voice-cloning prompts, joint alignment reduces word error rate from 3.18% to 2.42% at the reward-selected checkpoint (24.0% relative) and to 2.58% at the final checkpoint (19.0%), while WavLM speaker cosine rises from 0.733 to 0.748 and 0.755. Controlled experiments characterize recognition-identity trade-offs and the effects of corruption count, group composition, and weighting.

Adam is the default optimizer for training modern deep neural networks, yet its adaptive behavior remains poorly understood due to the complex interaction between its first- and second-moment exponential moving averages (EMAs). We study Adam in the tied-$β$ regime, where the two EMA decay rates are equal, and show that its adaptive dynamics can be expressed through a transformed ratio with approximately scale-stable behavior. Empirically, this transformed ratio exhibits a stable, heavy-tailed distribution across tasks, model scales, and training stages, in contrast to the variability of raw moment magnitudes. This empirical stability has both practical and conceptual consequences. First, we derive a recurrence for the transformed ratio, yielding a reparameterization of Adam that replaces the second moment with a compressible state. Leveraging its stable distribution, we show that a fixed 4-bit codebook is sufficient in our experiments to store this state without auxiliary scaling, achieving performance competitive with full-precision Adam. Second, the transformed ratio view clarifies Adam's connection to sign-based methods: Adam reduces to sign-based momentum modulated by the transformed ratio, and replacing it with a constant recovers Signum as a limiting case. This perspective further provides a simple rule for transferring learning rates between the two methods. Together, these results suggest that tied-$β$ Adam admits a simple and approximately stable ratio structure underlying its adaptive behavior and demonstrate its utility for both analysis and efficient implementation.

In this work, we propose a hybrid iterative deep Ritz method (H-IDRM) for a class of interface problems for second-order elliptic operators. It is based on a new mixed formulation of the problem and involves solving a sequence of convex minimization problems. We employ a level-set neural network architecture, featuring a level-set representation of the interface, to accommodate the piecewise smoothness of the solution and the flux. The approach involves only volumetric representations instead of duality pairing on the interface and avoids explicit interface sampling that is inconvenient for complex interface geometries. Further, we present an analysis of the method, including the errors arising from the neural network approximation, Monte Carlo approximation, iterative scheme, and penalty parameters. Numerical experiments indicate that the H-IDRM outperforms existing neural solvers on problems with high-dimensional domains, intricate interface geometries, and lower subdomain regularity.

Mesopic and low-light display transforms require, as their driving signal, a per-pixel scotopic-to-photopic luminance ratio (S/P); the exact spectral S/P is unavailable for ordinary RGB content, so a low-cost closed form that estimates S/P from a linear-RGB triplet is used in its place. Such closed forms exist but have been characterized only on narrowband / LED sources, i.e. spectrally sparse spectra, where a relative error of ~41% has been reported for a three-channel projection. Display content, however, is natural and broadband. We ask whether the same closed form is adequate there, using per-pixel spectral S/P from hyperspectral imagery as ground truth. On a daylight radiance time-series, a six-scalar closed form (three photopic and three scotopic channel weights) reproduces spectral S/P with a median error of ~0.07 that is time-invariant once the RGB input is chromatically adapted to D65; evaluated in un-adapted sRGB the error instead carries a color-temperature tilt across illuminants (~0.19), so adaptation is the enabling step for this use case. The result generalizes to an independent fifty-scene set (pooled median 0.024; 45/50 scenes within a pre-registered 0.10 band), with the few exceedances concentrated in saturated, spectrally-peaky surfaces that approach the narrowband regime (floral close-ups in this set). The scotopic weight vector is shown to be primary-model dependent, but the value used here is corroborated by a primary-free XYZ projection, and the median error stays within the band across all principled coefficient choices. We do not claim observer-validated appearance fidelity or adequacy on narrowband sources; both are out of scope. Both outcomes follow from the same three-channel projection: it is overwhelmed by spectrally sparse inputs and adequate on spectrally smooth ones.

Lepto-regression is defined as the machine learning process of constructing a Regression Tree of a target feature on itself. It is a novel, model-free method potentially revealing information on important sample structure properties. But it is yet not clear what the informational content of lepto-variance is and how it is related to other well-known statistics of a sample. One significant finding is that 58% of the historical US stock return variability is 1-bit lepto-variance that can not be explained by any financial factor. The central question investigated in this paper is to use small normal N(0, 1) drawn samples to explore how the 1-bit sample lepto-variance and lepto-ratio relate to sample variance, skewness and excess kurtosis. Using a large sample simulation, the lepto ratio of a normal is found to converge to 36.3%. For smaller normally distributed simulated N(0, 1) samples, while lepto-variance itself is highly correlated to sample variance, lepto-variance as a fraction of total variance is highly correlated to excess kurtosis. Both lepto-variance and lepto-ratio are orthogonal to sample skew. Another finding is that while lepto-ratio is strongly correlated to lepto-variance it remains orthogonal to sample variance.

Soft context compression condenses a context into a few memory tokens that a frozen LLM consumes in place of the raw text, but existing compressors fix the compression ratio at training and inference: each deployed ratio requires a separately trained model, and the chosen ratio is applied uniformly to all inputs, whose actual needs vary drastically. We propose FlexComp, a method-agnostic framework that decouples the ratio from both training and deployment: Matryoshka-style training samples the memory budget $K$ per instance, turning one model into an any-ratio compressor, and the budget is then chosen per input by: (1) confidence-based cascade routing or (2) a lightweight learned $K$ predictor. Across ICAE, 500xCompressor, and SAC on MRQA, a single FlexComp model matches separately trained fixed-ratio specialists with minimal degradation. Cascade routing preserves over 98% of the mildest ratio's accuracy at up to 266x average compression; the $K$ predictor, in a single compression-decoding pass, reaches 158-236x within 0.7 F1 of the mildest ratio. At serving-scale batch sizes, the $K$ predictor cuts context KV cache by 50% and improves decoding throughput by 47%.

Thinking Mode Fusion (TMF) enables large language models to support both concise responses and long-form reasoning by unifying a non-thinking mode and a thinking mode within a single model. However, its training dynamics, including the \emph{data ratio} and \emph{training schedule} between the two modes, remain underexplored. In this work, we present a systematic study of TMF by analyzing the effects of the training schedule and data ratio between thinking and non-thinking modes. Focusing on mathematical problem solving, we construct a benchmark with multiple thinking-to-non-thinking data ratios and three training schedules. Our results reveal an asymmetric interaction between the two modes: increasing the ratio of non-thinking supervision reduces the accuracy of the thinking mode. We further show that different training schedules modulate this trade-off and that the optimal schedule depends on the data ratio. Finally, we quantify a negative correlation between non-thinking and thinking mode supervision, highlighting an inherent tension between these two modes. These findings provide practical guidance for designing effective TMF training settings. All code and data are released to support further research at: \href{https://github.com/caocongfeng/Fusion-Bench.git}{\textbf{Fusion Bench}}.

In fixed-budget best-arm identification, also known as ranking and selection, an algorithm has a sampling budget to distribute across $K$ arms. Each sample provides noisy feedback about that arm's mean, and the goal is to identify the arm with the largest mean. A common performance benchmark is the static oracle: a non-adaptive strategy that knows the means in advance and chooses fixed sampling proportions to maximize the exponential decay rate of the probability of incorrect identification. Several adaptive algorithms have been constructed such that their sampling proportions converge to the static oracle proportions. However, it has remained open whether any algorithm could match the static oracle's error decay rate uniformly across all problem instances. We answer this in the negative. For any $K\ge 3$ and for rewards drawn from any one-parameter natural exponential family, we show that for any algorithm, there is at least one instance where the error decay rate is at most $\left(1 + \frac{\log(K)}{8}\right)^{-1}$ times that of the static oracle. This also answers the open question posed by Qin (2022), showing that fixed-budget best-arm identification does not admit a complexity.

Authorship verification (AV) is the task of determining whether two texts were written by the same author. In a forensic context, the strength of AV evidence can be quantified using likelihood ratios. Most AV methods are score-based and deriving well-calibrated likelihood ratios from these scores requires a separate calibration model. This, in turn, requires additional amounts of case-relevant data, which is often time-consuming to obtain and prepare. This study proposes two novel normalisation techniques, the Square Root Correction and the Hapax Correction, for deriving likelihood ratios from the AV method LambdaG without the need of a calibration model (Nini et al. 2026). These corrections are designed to mitigate the overestimation of evidential strength that may result from long or highly repetitive texts. Performance is evaluated against logistic regression calibration across fifteen corpora and a range of text lengths (100-9,500 tokens), using the log-likelihood ratio cost (Cllr). The proposed methods achieve performance comparable to logistic regression calibration, with the Hapax Correction outperforming it in approximately 45% of tests (weighted by corpora). Furthermore, performance was more frequently close (within 5%) when the Hapax Correction was outperformed by logistic regression calibration, compared with the reverse comparison. Eliminating the need to train a calibration model reduces data-requirements, time and complexity, thereby increasing the accessibility and transparency of forensic text comparison. This combination of empirical performance and practical advantages supports the adoption of the proposed methods in forensic settings.

Analogical proportions link four items a, b, c, d by a relation stating that ``a is to b as c is to d", a, b, c, d being the formal representation of real world entities, ranging from simple numerical values to more complex structures such as profiles. Accordingly, $a, b, c, d$ could be atomic values like Boolean, nominal or numerical values, more generally vectors of such values, or even families of items represented by logical formulas. In this paper, we consider another representation setting, which is the probabilistic one. Precisely, the article proposes a study of {analogical} proportions between probabilities, whether they are simply between probability values, or between distributions (which requires the preservation of their normalization). More particularly, we study the properties of definitions based on arithmetic proportion, or on a combination of the former with geometric proportion, while other options are also discussed. Previous works have shown that when four profiles a, b, c, d, represented as vectors, form analogical proportions componentwise, it is likely that their classes form an analogical proportion also. This is the basis of an analogical proportion-based classification method that can produce accurate predictions. Similarly, in this paper, each profile is associated with a distribution describing the frequencies of the possible values of a discrete attribute of interest. We then discuss and experimentally investigate if the distributions associated to four profiles forming an analogical proportion themselves also form an analogical proportion.

We study the log-alignment ratio (LAR), a measure of parameter-activation alignment, introduced in parameterization theory. We reformulate it as the overlap between a weight spectrum $p$ of the normalized squared singular values of a matrix and an activation spectrum $q$ of the normalized squared projections of inputs onto its singular directions. We show that unembedding LAR tracks the transition between memorization and generalization in two different settings by capturing the spread of $p$ and $q$ during training. In grokking, LAR predicts the effective dimension of the learned function: $k \approx n^{2(1-\text{LAR})}$, where $n$ is the input dimension of the matrix. In 3B-parameter language model pre-training, its deviation from a non-overfitting baseline tracks the generalization gap, and its rate of decline increases as overfitting approaches. LAR is computable from quantities available during the forward pass with negligible computational overhead, and requires no held-out validation data.

Understanding how LLMs reason is hindered by a practical asymmetry: while their generated outputs are observable, the underlying reasoning patterns remain opaque. Relying on single probes, such as Mutual Information Peak (MIP) or Deep-Thinking Ratio (DTR), risks underestimating the genuine inferential structure. To response this deficiency, we present an Integrated, cross-Architecture Reasoning (IAR) framework, designed to provide a unified approach to LLM reasoning interpretability. Specifically, we first propose to use bandwidth-calibrated MIP coupled with Tukey IQR peak-detection to isolate reasoning-crucial tokens at the output layer. Second, we performed an overlap analysis between MIP-picked tokens and DTR-deep tokens to trace the cross-layer trajectories of those tokens. This also discloses whether reasoning-crucial tokens are computation-intensive as well, further facilitating to understand how reasoning patterns evolve across model layers. Finally, we apply a Jaccard stability metric over multi-domain problems to verify if the MIP-identified tokens are reasoning quality-guaranteed. Extensive experiments on six models across four domains (mathematics, code, logic, and common sense) demonstrate IAR's generalizable interpretation capabilities across architectures. Our code is available at https://github.com/LeonardoMatthew/IAR.

When treatment effects are naturally expressed as ratios -- as in medicine, pricing, and marketing -- the ratio-based CATE $τ(x) = E[Y|W=1,X=x] / E[Y|W=0,X=x]$ is the appropriate estimand. Yet existing estimators either impose a log-linear parametric structure or apply generic regression without robustness guarantees for this functional. We introduce the Q-Learner, which decomposes $τ(x)$ into a product of two odds ratios, reducing ratio-CATE estimation for binary outcomes to two propensity classification tasks. We further derive doubly robust augmentations for both S/T- and Q-style ratio learners and characterize their distinct robustness properties. In benchmarks on seven RCT datasets, the Q-Learner is the most consistently competitive method in low-conversion regimes, where its propensity-only construction sidesteps the imbalanced regression that hurts outcome-based estimators. On four observational datasets, where propensity must be estimated and confounding cannot be ruled out, the DR learners introduced here decisively come out on top, making them practitioners' natural default for confounded observational data.

We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object type and configuration, we establish an explicit cross-ratio formula, prove that it recovers the appropriate classical invariant, and identify the canonical pairwise measurement operator. A systematic duality analysis further revealed that all eight configurations organize into four dual pairs under the Hodge dual, and that all measurement operators reduce to either the commutator or the commutator dual, depending solely on the geometric configuration rather than on object grade. In each case the formula recovers the appropriate classical invariant: signed distance ratios for parallel configurations and sine cross-ratios for secant ones. These results establish the cross-ratio as a grade-agnostic projective invariant within PGA, and provide a constructive foundation for defining n-dimensional homographies directly from prescribed invariants.

Large language models can follow complex instructions in a single turn, yet over long multi-turn interactions they often lose the thread of instructions, persona, and rules. This degradation has been measured behaviorally but not mechanistically explained. We propose a channel-transition account: goal-defining tokens become less accessible through attention, while goal-related information may persist in residual representations. We introduce the Goal Accessibility Ratio (GAR), measuring attention from generated tokens to task-defining goal tokens, and combine it with sliding-window ablations and residual-stream probes. When attention to instructions closes, what survives reveals architecture. Across architectures, the transition yields qualitatively distinct failure modes: some models preserve goal-conditioned behavior at vanishing attention, others fail despite decodable residual goal information, and the layer at which this encoding emerges varies from 2 to 27. A within-model causal ablation that force-closes the attention channel in Mistral collapses recall from near-perfect to 11% on a 20-fact retention task and raises persona-constraint violations above an adversarial-pressure baseline without user pressure, with both effects emerging at the predictable crossover turn. Linear probes recover per-episode recall outcomes from residual representations with AUC up to 0.99 across all four primary architectures, while input embeddings remain at chance. Across architectures and model scales, the gap between attention loss and residual decodability predicts whether goal-conditioned behavior survives channel closure. We contribute GAR as a diagnostic, the channel-transition framework as a controlled mechanistic account, and a parametric prediction of failure timing under windowed attention closure.

Modern vision models achieve remarkable accuracy, but explaining where evidence arises, what the model encodes, and how internal computations assemble that evidence remains fragmented. We introduce an iERF-centric framework that unifies local, global, and mechanistic interpretability around a single analysis unit: the pointwise feature vector (PFV) paired with its instance-specific Effective Receptive Field (iERF). On the local side, Sharing Ratio Decomposition (SRD) expresses each PFV as a mixture of upstream PFVs via sharing ratios and propagates iERFs to construct class-discriminative saliency maps. SRD yields high-resolution, activation-faithful explanations, is robust to targeted manipulation and noise, and remains activation-agnostic across common nonlinearities. For the global view, we introduce Concept-Anchored Feature Explanation (CAFE), which utilizes the iERF as a semantic label, grounding abstract latent vectors in verifiable pixel-level evidence. With CAFE, we address the challenge of non-localized sparse autoencoder latents--especially in Transformers, where early self-attention mixes distant context. To answer how representations are composed through depth, we propose the Interlayer Concept Graph with Interlayer Concept Attribution (ICAT), which quantifies concept-to-concept influence while isolating layer pairs; an interlayer insertion, deletion protocol identifies Integrated Gradients as the most faithful instantiation. Empirically, across ResNet50, VGG16, and ViTs, our framework outperforms baselines in both fidelity and robustness, successfully interprets dispersed SAE features, and exposes dominant concept routes in correct, misclassified, and adversarial cases. Grounded in iERFs, our approach provides a coherent, evidence-backed map from pixels to concepts to decisions.

The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods. We propose a new neural operator approach that bridges classical convex-concave splitting schemes with physics-informed learning to accelerate the simulation of phase-field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase-field model. By training the neural operator with an energy-splitting variational formulation, we enforce the energy dissipation property of the underlying models. We further introduce a custom Reaction-Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen-Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically-informed training provides better generalization in out-of-distribution evaluations than data-driven training, while achieving faster inference than traditional Fourier spectral methods.

Large language models (LLMs) have demonstrated impressive reasoning capabilities by scaling test-time compute via long Chain-of-Thought (CoT). However, recent findings suggest that raw token counts are unreliable proxies for reasoning quality: increased generation length does not consistently correlate with accuracy and may instead signal "overthinking," leading to performance degradation. In this work, we quantify inference-time effort by identifying deep-thinking tokens -- tokens where internal predictions undergo significant revisions in deeper model layers prior to convergence. Across four challenging mathematical and scientific benchmarks (AIME 24/25, HMMT 25, and GPQA-diamond) and a diverse set of reasoning-focused models (GPT-OSS, DeepSeek-R1, and Qwen3), we show that deep-thinking ratio (the proportion of deep-thinking tokens in a generated sequence) exhibits a robust and consistently positive correlation with accuracy, substantially outperforming both length-based and confidence-based baselines. Leveraging this insight, we introduce Think@n, a test-time scaling strategy that prioritizes samples with high deep-thinking ratios. We demonstrate that Think@n matches or exceeds standard self-consistency performance while significantly reducing inference costs by enabling the early rejection of unpromising generations based on short prefixes.