--- title: 'Input Time Scaling: Concepts & Applications' url: https://www.emergentmind.com/topics/input-time-scaling type: topic --- # Input Time Scaling: Concepts & Applications to=arxiv_search.query 大发快三怎么json anasiyana 1: {"query":"all:\"Input Time Scaling\" OR ti:\"Input Time Scaling\" OR abs:\"Input Time Scaling\"","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} 天天彩票软件сәтာքներ to=arxiv_search.query code ￣色 none to=arxiv_search.query 时时彩后 to=arxiv_search.query code 大发游戏官网 none to=arxiv_search.search 大发彩票官网 to=arxiv_search.search code 全民彩票天天送json {"query":"Input Time Scaling","max_results":5} Input time scaling is a context-dependent term used across several technical literatures to denote methods that treat temporal structure, temporal resolution, or input-side query transformation as an explicit design variable rather than a fixed background parameter. In recent work on agents and large language models, it denotes allocating resources to query refinement and strategy-conditioned inputs during both training and testing [2508.13654]. In text-to-image generation, it appears as input-side inference-time scaling through prompt rewriting while the image backbone remains frozen [2510.12041]. In time-series forecasting, it refers to scaling the look-back horizon \(H\) and analyzing its interaction with dataset size \(D\) and model size \(N\) [2405.15124]. In time and frequency metrology, it denotes bringing heterogeneous clock signals to a common time representation and digitally steering a composite oscillator [2307.03751], including optical-clock steering of a real-time time scale [1803.03485]. In quantum control, it is the explicit choice of a rescaled time variable \(s(t)\) or \(\Lambda(t)\) that re-times a reference evolution [2208.14285, 2106.05155]. A related financial usage redefines the time axis itself so that return distributions obey simple scaling with \(H=1/2\) [1612.07802]. Taken together, these works indicate that no single field-independent definition is in use. ## 1. Terminological scope and unifying pattern A useful common denominator is that input time scaling promotes some temporal quantity to a first-class control parameter. Depending on the field, that quantity may be a query transformation budget, a look-back horizon, a local clock representation, a rescaled Schrödinger time, a timestep \(h\), or a continuous dilation field \(a({\bf x},t)\). The review literature on inference-time scaling makes this distinction explicit by separating output-focused methods from input-focused methods; the latter encompass few-shot prompting and retrieval-augmented generation, where additional inference compute is spent on what the model receives as input rather than on search over outputs [2510.10787]. This broader pattern is not merely terminological. In each domain, the temporal variable being scaled mediates a trade-off between information and cost. Longer or richer inputs can improve reasoning, but they also change the effective state space seen by the model. Larger look-back windows reduce Bayesian forecasting error but can increase approximation error. Slowing or accelerating a quantum trajectory can preserve a target path while redistributing control effort. Enlarging a physical driver timescale can preserve similarity only if geometry, losses, and energy are co-scaled. The recurrent theme is therefore not “more time is better,” but “time-like inputs become optimization variables with nontrivial constraints.” ## 2. Query-side scaling in language models and multimodal generation In the agent and LLM literature, “Input Time Scaling” is introduced as a scaling paradigm complementary to data and training scaling and to inference-time scaling. The core operation is an input transformation \[ \tilde{x} = \phi(x; s), \] where \(s \in \{N,S,U,R\}\) denotes No-Persona, Persona-Similar, Persona-Unsimilar, or Persona-Random query construction [2508.13654]. The paper reports a training-testing co-design phenomenon: query strategies must be applied during both training and testing, and only applying strategies on training or testing seriously degrades performance. It also reports results that run against the usual “garbage in, garbage out” inductive bias: adding irrelevant information to queries and randomly selecting examples from a minimally filtered dataset can perform best, while increasing data size from \(1\text{k}\) to \(15\text{k}\) with similar quality can perform worse. With models trained on Qwen2.5-32B-Instruct, the reported pass@1 results reach AIME24 \(76.7\%\) and AIME25 \(76.7\%\), and a majority vote of three models reaches AIME24 \(76.7\%\) and AIME25 \(80\%\) [2508.13654]. A closely related multimodal instantiation appears in text-to-image generation as “input-side inference-time scaling.” There, the scaled object is not a reasoning trace but the prompt itself. A prompt rewriter LLM produces refined prompts, a frozen text-to-image backbone renders them, and multimodal judges assign pairwise preferences over image quality, general image-text alignment, physical image-text alignment, and aesthetics; these preferences are then used in an iterative DPO pipeline [2510.12041]. The method is model-agnostic and transfers across FLUX, Stable Diffusion 3.5, and JanusPro backbones. Representative results include FLUX.1-schnell on Pick-a-Pic v2 improving from average \(0.401\) without rewriting to \(0.510\) with the general rewriter and \(0.579\) with the aesthetics rewriter, and FLUX.1-dev on GenEval improving overall from \(0.70\) to \(0.79\) with the general rewriter [2510.12041]. The input-focused review literature places these methods within a larger inference-time taxonomy. Few-shot prompting, query expansion, retrieval, reranking, and multimodal context construction are all treated as ways of scaling inference by enriching the input channel instead of altering model weights [2510.10787]. This suggests a general interpretation of query-side input time scaling: inference budget is spent on constructing better conditional contexts. ## 3. Horizon scaling in time-series forecasting In time-series forecasting, input time scaling refers to scaling the look-back horizon \(H\). The central theory treats forecasting loss as \[ L = L_{Bayesian} + L_{approx}, \] with dataset size \(D\), model size \(N\), and input horizon \(H\) defined in an intrinsic space whose dimension \(d_I(H)\) grows linearly with \(H\) [2405.15124]. The Bayesian term decreases as the horizon grows, because longer history provides more information. The approximation term grows with the effective intrinsic dimension, because longer horizons enlarge the space that finite data and finite-capacity models must cover. The paper formalizes a phase boundary with \[ \xi = \frac{D}{NH}, \] which separates data-dense and data-sparse regimes [2405.15124]. This formulation explains why “more past is not always better.” The paper predicts a finite optimal horizon \(H^*\), and the empirical results show U-shaped loss-versus-horizon curves. For fixed model and varying amounts of training data, loss first falls and then rises as \(H\) grows. The optimal horizon increases with dataset size. Feature degradation also matters: for Exchange, the optimal horizon is reported as \(<30\) even using \(100\%\) of training data, while for ETTh1 it is \(>300\) even using only \(11\%\) of training data [2405.15124]. The significance of this usage is methodological. In contrast to scaling laws in language or vision, where larger context or resolution is often beneficial, the forecasting literature treats horizon as a third scaling axis with its own optimum. Input time scaling here means choosing \(H\) jointly with \(D\) and \(N\), not maximizing it by default. ## 4. Metrological time scales and composite clocks In time and frequency metrology, a time scale is a realizable continuous time function built by combining several physical clocks, and a composite clock is the physical and algorithmic realization of that time scale. The digital approach described in “The Digital Revolution, also for Time Scales” measures each input clock against a local oscillator, converts phases to a common phase-time representation, removes deterministic drift, performs weighted averaging, \[ T_{\text{comp}}(t_k) = \sum_{i=1}^{N} w_i(t_k)\,T'_i(t_k), \] and then phase-locks the local oscillator to the computed composite phase [2307.03751]. In this setting, input time scaling means ingesting multiple clocks at arbitrary frequencies, bringing them to a common time/frequency representation, digitally steering and combining them, and mapping the result back into one physical output oscillator. A preliminary experiment composing three active hydrogen masers reports a stability at \(1\) s of \[ \sigma_y(1\,\text{s}) \approx 4.5 \times 10^{-14}, \] compatible with the short-term stability of the average of the masers [2307.03751]. A related but distinct metrological usage appears in the optical-clock time-scale literature. “Months-long real-time generation of a time scale based on an optical clock” describes TA(Sr), an optically steered time scale generated continuously for half a year by steering a hydrogen maser with intermittent operation of a \(^{87}\)Sr optical lattice clock [1803.03485]. Here the input is sparse but high-accuracy optical-clock data; the system estimates the maser drift in a moving window and applies steering corrections so that the integral of the corrected fractional frequency is zero over each update interval. The reported variation of TA(Sr) \(-\) TT(BIPM16) is \(0.79\) ns after \(5\) months, and the time scale is reported as stable as TAI with accuracy at the \(10^{-16}\) level [1803.03485]. In both cases, the conceptual shift is from manipulating RF waveforms directly to manipulating time/phase information. Input time scaling therefore denotes a representational normalization problem: diverse clocks are first reduced to digital phase or time streams, then recombined algorithmically. ## 5. Quantum time rescaling and re-timed dynamics In quantum control, input time scaling is the explicit choice of a rescaled time \(s(t)\), or equivalently \(\alpha(t)=ds/dt\), that prescribes how a reference evolution is replayed in laboratory time. In the fast-forward theory for nonadiabatic transitions, the target condition is that populations at laboratory time \(t\) match those of the reference dynamics at rescaled time \(s(t)\), \[ |\langle \sigma|\Psi_{\text{FF}}(t)\rangle|^2 = |\langle \sigma|\Psi_{\text{ref}}(s(t))\rangle|^2, \] and the corresponding Hamiltonian takes the form \[ \hat{H}_{\text{FF}}(t)=\hat{H}_{\text{ref}}(s)+\frac{ds}{dt}\big[\hat{H}_{\text{cd}}(s)+\hat{H}_{\text{nad}}(t)\big], \] where \(\hat{H}_{\text{cd}}\) is the counterdiabatic term and \(\hat{H}_{\text{nad}}\) is a nonadiabatic reproduction term [2208.14285]. The method can speed up, slow down, pause, or rewind a reference dynamics. In the adiabatic limit, the nonadiabatic reproduction term averages out and the construction reduces to shortcuts to adiabaticity. The non-Hermitian counterpart rescales time by a monotone map \(\Lambda(t)\), with transformed Hamiltonian \[ H_\Lambda(t)=\dot{\Lambda}(t)\,H(\Lambda(t)), \] and uses that freedom to reduce norm loss in open-system state transfer without adding new couplings [2106.05155]. In the two-level and three-level examples studied there, appropriate time scaling improves fidelity while preserving the ratio between actual speed and the non-Hermitian quantum speed limit. The paper derives the bound \[ \dot{\phi}(t) \le \frac{\chi(t)}{\hbar}, \] with \(\chi\) containing both Hermitian and anti-Hermitian contributions, and shows that suitably scaled driving preserves optimality with respect to this bound while significantly reducing damping of the quantum state norm [2106.05155]. This usage is conceptually precise: the input is literally a time map. What is optimized is not only the control field but the schedule by which the reference trajectory is traversed. ## 6. Time scaling in optimal control and multiscale simulation In optimal control, time scaling enters when the discretization step \(h\) becomes a decision variable. For linear dynamics discretized as \[ x_{k+1}=x_k+h(Ax_k+Bu_k), \] the products \(h x_k\) and \(h u_k\) make the problem nonconvex [2504.13170]. The proposed semidefinite relaxation introduces \[ y=\begin{bmatrix}1 & h & r & hr\end{bmatrix}, \qquad Y=\frac{1}{h}yy^\top, \] selects only the bilinear terms that actually appear in the time-scaled dynamics and cost, and augments the formulation with tightening equalities and inequalities. The method extends to piecewise-affine systems by representing mode sequences as paths in a graph of convex sets, so that the time-scaled PWA optimal-control problem can be solved through a single semidefinite program [2504.13170]. A different but allied formulation appears in multiscale timestepping for astrophysical simulation. There the governing equation \[ \frac{D{\bf U}_i}{Dt}=\mathcal{F}({\bf U}_j,\ldots) \] is replaced by \[ \frac{1}{a_i}\frac{D{\bf U}_i}{Dt}=\mathcal{F}({\bf U}_j,\ldots), \] where \(a_i=a({\bf x}_i,t,\ldots)\) is a continuous dilation factor with \(0