Compressor-Predictor Systems

• Compressor-Predictor Systems are frameworks that condense raw, high-dimensional data into compressed representations for targeted inference.
• They employ staged pipelines and rate–distortion analysis to balance mutual information preservation with computational efficiency.
• These systems are applied across language models, time series forecasting, and embedded sensing to optimize predictive maintenance and control.

A compressor–^^^^1^^^^ system is a broad architectural and methodological paradigm where a "compressor" module distills raw, high-dimensional, redundant, or temporally extended data into a compressed intermediate representation, which is then consumed by a "predictor" module tasked with producing decisions, forecasts, reconstructions, or other inferences. This architecture recurs in machine learning, control, signal processing, scientific data analysis, and industrial systems engineering, generally enabling more efficient computation, lower resource requirements, and potential gains in accuracy or interpretability.

1. Formal Taxonomy and General Principles

The canonical compressor–predictor workflow is a staged pipeline:

1. Compression: The input $X$ (which may be raw text, multichannel time series, spatial arrays, sensor streams, or other structured data) is transformed into a compressed representation $Z$ by a mapping $p(z|x)$, often designed to retain only information relevant for downstream prediction.
2. Prediction: A predictor processes $Z$ to output $Y$ (e.g., a label, answer, predicted future, or reconstructed signal), typically as $p(y|z)$.

This is abstracted as: $$X \;\xrightarrow{\ p(z\mid x)\ }\; Z \;\xrightarrow{\ p(y\mid z)\ }\; Y.$$ Performance is measured by end-to-end accuracy, reconstruction fidelity, or application-specific metrics. The mutual information $I(X;Z)$ quantifies the amount of task-relevant information preserved through compression, providing a task-agnostic, information-theoretic foundation for evaluating and designing such systems (He et al., 25 Dec 2025).

2. Mathematical and Information-Theoretic Foundations

Mutual Information and Rate–Distortion

In contemporary LLM systems, the compressor can be viewed as a noisy channel, and $I(X;Z)$ acts as a key bottleneck metric. Empirically, increasing compressor size tightly correlates with both higher mutual information and improved downstream task performance, while making compression more concise in bits or tokens per unit information (He et al., 25 Dec 2025). The rate–distortion notion is formalized as: $$R = \frac{I(X;Z)}{\mathbb{E}[\#\mathrm{tokens}(Z)]}, \quad D = 1 - \mathrm{accuracy}.$$ Observed rate–distortion curves in LLM compressor–predictor systems follow an exponential shape: $D(R) \approx C e^{-b\,R} + D_0,$ with $D_0$ the residual floor set by model or data intrinsic limitations.

A similar information-theoretic analysis applies in the compressed observation learning setting, where the conditional distortion–rate function

$$D_{Y|X}(R,P) = \inf_{p(\widehat{Y}\mid X,Y): I(Y;\widehat{Y}|X)\leq R} \mathbb{E}[\ell(Y,\widehat{Y})]$$

characterizes the minimum achievable loss when only a compressed version of $Y$ is available for statistical learning, possibly with side information $X$ (0704.0671).

Predictive Modeling of Compression Performance

Both black-box and analytical predictor models can anticipate the effects of different compressor choices, compression parameters, and (in lossy settings) error bounds, on post-compression data utility. Statistical predictors based on quantized entropy, spatial correlation, and linear or non-parametric regression achieve median percentage prediction errors below 12% for scientific data (Underwood et al., 2023), and analytical entropy-residual models allow precise ratio–quality trade-off prediction in error-bounded lossy compressors (Jin et al., 2021).

3. Architectures Across Domains

LLMs and Agentic Systems

Agentic LLM workflows commonly compose a local, smaller "compressor" model—summarizing a long context or history—feeding into a larger predictor LLM to answer queries with limited available context. Mutual information between context and compressed summary is the most reliable predictor of overall system quality, superseding traditional heuristic metrics such as summary length or perplexity. Notably, scaling the compressor, not the predictor, most efficiently raises system accuracy and token efficiency (He et al., 25 Dec 2025).

Time Series and Scientific Data

In scientific and industrial scenarios, compressor–predictor systems enable:

• Predictive ratio–quality modeling for error-bounded lossy compression, optimized through small-sample entropy/statistics and mapping to rate and distortion (Jin et al., 2021, Underwood et al., 2023).
• Predictability–aware compression of multichannel time series, where compression is done via orthogonal circulant key matrices ("PCDF"), yielding single-channel surrogates that retain cross-channel dependencies and enable faster, more scalable prediction (Liu et al., 31 May 2025).
• End-to-end pipelines for predictive maintenance and anomaly detection in compressor-based machines, with the compressor serving to extract stationary or low-dimensional representations for downstream ML/DL-based predictors (e.g., LSTM, CNN, hybrid autoencoders), and explicit modeling of temporal segments, quantization, and statistical properties for fault and change-point detection (Forbicini et al., 2024, Łobodziński, 2024).

Control and Optimization

In physical systems engineering, compressor–predictor patterns arise in:

• Model predictive control (MPC) of gas pipeline networks actuated by compressors, where nonlinear system dynamics are replaced by linearized predictors that approximate the behavior with provable stability and error bounds, effecting real-time feedback control under computational constraints (Baker et al., 2023).
• Real-time surge prediction and adaptive PD control for compressor stability using reduced-order models and state-space predictors (Hosseindokht, 6 Mar 2025).

Embedded Sensing

On-chip compressor–predictor modules, such as lossless slope-prediction and dynamic coding in wireless ECG sensors, reduce data rates and memory/energy footprint, while preserving the information required by downstream classifier or reconstruction algorithms (Deepu et al., 2014).

4. Methods for Learning and Designing Compressor–Predictor Pipelines

A selection of evidence-based methods:

Approach	Compression	Prediction / Learning
Monte-Carlo MI estimation (LLMs) (He et al., 25 Dec 2025)	Stochastic sequence	Cross-entropy/perplexity metric
Ratio–quality modeling (Jin et al., 2021)	Entropy histograms	Closed-form bit-rate, PSNR, SSIM
Black-box regression (Underwood et al., 2023)	Quantized entropy/stats	Linear/spline models for ratio
Predictability-aware compression (Liu et al., 31 May 2025)	Circulant key matrices	Standard single-channel forecaster
Supervised/unsupervised predictive maintenance (Łobodziński, 2024)	LPPL model fit	Trend/extrema analysis

In all these methods, explicit feature extraction, dimensionality reduction, quantization, and entropy estimation serve as compressor building blocks, often in conjunction with application-specific predictors (statistical models, deep networks, or analytic control laws).

5. Quantitative Evidence and Trade-Off Analysis

Key empirical findings underline the nuanced trade-offs in compressor–predictor design:

• In LLM-based pipelines, scaling compressor size from 1.5B to 7B parameters achieves 1.6$\times$ higher accuracy, 4.6$\times$ greater conciseness, and 5.5$\times$ mutual information per token; scaling the predictor provides only marginal gains (He et al., 25 Dec 2025).
• Predictability-aware time series compression (PCDF) yields 2$\times$–10$\times$ speedup in inference runtime while preserving mean squared error across diverse forecasting models and datasets; best Cobb–Douglas aggregate (error$\times$runtime) is achieved in 85% of tested scenarios (Liu et al., 31 May 2025).
• For error-bounded lossy scientific compression, hybrid ratio–quality models reach $\approx$95% bit-rate accuracy and $\approx$97% PSNR accuracy, reducing tuning time by up to 18.7$\times$ and enabling 3.4$\times$ faster I/O (Jin et al., 2021).
• Supervised FP/FD and forecasting in compressor-based machines: 1D-CNNs and LSTM autoencoders outperform classical ML and statistical baselines, but require careful handling of class imbalance and domain adaptation; accuracy/precision above 90% is typical when adequate data is available (Forbicini et al., 2024).

6. Practical Guidelines, Limitations, and Future Directions

Design and deployment guidelines include:

• Prioritize compressor scaling and bit-efficiency; mutual information per output unit is the most robust task-agnostic proxy (He et al., 25 Dec 2025).
• Use lightweight, compressor-agnostic statistical predictors (e.g., entropy, spatial correlation) to automate compressor selection and parameter tuning (Underwood et al., 2023).
• Augment black-box pipelines with closed-form or sample-based analytical modeling to replace brute-force search across error bounds and predictors (Jin et al., 2021).
• In edge/cloud scenarios, integrate orthogonal-key compressive schemes for multichannel streams to enable single-predictor architectures with reduced computational burden (Liu et al., 31 May 2025).
• When extendibility or transfer is needed, prefer modular systems whose compressor and predictor blocks can be independently retrained or replaced.
• For real-time control, ensure the predictor's linearization errors remain provably bounded via Lyapunov-based analysis to justify the use of simplified models (Baker et al., 2023, Hosseindokht, 6 Mar 2025).

Open limitations include dependence on the compatibility of compressor and predictor types, domain shifts requiring retraining of predictive models, and challenges in bridging extreme compression ratios without instability. A plausible implication is that hybrid approaches, physics-informed compression, and foundation models for temporal data promise to further enhance compressor–predictor systems by bridging gaps between interpretability, efficiency, and generalization (Forbicini et al., 2024).

7. Applications and Impact Across Disciplines

Compressor–predictor systems have significant impact in:

• Large-model question answering and research assistants, where local compressors extend effective context length for cloud-scale LMs at reduced cost (He et al., 25 Dec 2025).
• Scientific data management, enabling rapid tuning and compression/analysis pipelines without repeated full compression runs (Underwood et al., 2023, Jin et al., 2021).
• Industrial time series forecasting and predictive maintenance, where unsupervised (LPPL-based) and supervised (DL-based) pipelines achieve high-precision fault prediction and system health monitoring (Łobodziński, 2024, Forbicini et al., 2024).
• Embedded medical sensing, where ultra-low-power on-chip compressors enable long-duration wireless monitoring without sacrificing diagnostic quality (Deepu et al., 2014).
• Large-scale pipeline networks and compressor actuation in energy systems, supporting optimal control and stability via coupled model linearization and real-time feedback (Baker et al., 2023).
• High-dimensional statistical inference, as in Bayesian compressed regression, where random projections enable scalable, near-parametric learning in $p\gg n$ regimes (Guhaniyogi et al., 2013).

These examples highlight the flexibility and centrality of compressor–predictor frameworks in contemporary computational, engineering, and data science ecosystems.