Energy Conserving Subsampling Methods

• Energy conserving subsampling is defined as methods that reduce redundant samples while preserving a system-specific energy invariant, ensuring both computational efficiency and quality retention.
• It is applied in diverse fields such as Hamiltonian Monte Carlo for high acceptance rates, video decoding for lower energy consumption, and sensor networks to extend device lifetimes.
• By matching the subsample in both processing and evaluation steps, these techniques achieve significant speedups and energy savings without compromising task performance or perceptual quality.

Energy conserving subsampling denotes a family of methods that reduce the number of processed samples while preserving a system-specific energy-related invariant. In a narrow sense, the term is identified with subsampling schemes for Hamiltonian Monte Carlo (HMC) that preserve the Hamiltonian used in proposal generation and acceptance. In a broader engineering sense, it describes subsampling, downscaling, or selective readout strategies that reduce decoding, rendering, sensing, acquisition, or communication energy while maintaining perceptual quality or task performance in video coding, graphics, sensor networks, and embedded vision (Dang et al., 2017, Herglotz et al., 2023, Premkumar et al., 5 Feb 2026, Anglada et al., 2022, Majumdar et al., 2019, Iqbal et al., 2021).

1. Scope and unifying interpretation

In current literature, the concept appears in two distinct but related forms. In Bayesian computation, the conserved quantity is explicit: HMC relies on approximate conservation of a Hamiltonian, and subsampling is admissible only if the same subsample determines both the simulated dynamics and the Metropolis acceptance probability. In media and sensing systems, the “energy-conserving” aspect refers instead to reducing physical energy expenditure by avoiding redundant samples, frames, fragments, chroma components, or sensor readings while preserving a quality criterion such as GREED, ColorVideoVDP, PSNR, SSIM, RMSE, AUC, or FPS (Dang et al., 2017, Herglotz et al., 2023, Premkumar et al., 5 Feb 2026, Anglada et al., 2022, Majumdar et al., 2019, Iqbal et al., 2021).

Domain	Subsampled object	Preserved criterion
HMC	Data subset $S$	Modified Hamiltonian $H_S$ and high acceptance
Video coding	Frames and spatiotemporal frequency bands	ST-GREED quality at lower decoding energy
Adaptive coding	Resolution/chroma representations	ColorVideoVDP at lower decoding time
GPU rendering	Tile shading samples	PSNR/SSIM targets
Sensor networks	Active sensor subset	Low-RMSE field reconstruction
ROI tracking	Sensor pixels outside predicted ROI	AUC/FPS under lower power

A plausible unifying interpretation is that energy conserving subsampling is effective only when the discarded samples are redundant with respect to a structural prior already exploited by the system: posterior smoothness and control variates in HMC, low temporal activity in video, low local spatial frequency in graphics, low-rank spatial fields in sensor networks, or predictable ROI motion in tracking.

2. Hamiltonian Monte Carlo with energy-conserving subsampling

The narrowest and most explicit usage is the HMC-ECS method of “Hamiltonian Monte Carlo with Energy Conserving Subsampling” (Dang et al., 2017). Standard HMC introduces parameters $\theta \in \mathbb{R}^d$, momentum $p \in \mathbb{R}^d$, and Hamiltonian

$H(\theta,p)=U(\theta)+K(p),$

with $U(\theta)=-\log \pi(\theta)$ and $K(p)=\tfrac12 p^T M^{-1}p$. Continuous dynamics conserve $H$ exactly, which is the basis for distant proposals with high acceptance.

The central difficulty is that in large-$n$ settings, evaluating

$U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$

and its gradient at each leapfrog step is expensive. Naive data subsampling breaks HMC because the approximate Hamiltonian used in the integrator differs from the one used in the accept/reject step. The resulting energy mismatch destroys the conservation property and leads to very low acceptance rates, especially in high dimensions.

HMC-ECS resolves this by augmenting the state to $H_S$0, where $H_S$1 indexes a random data subsample of size $H_S$2, and alternating two Gibbs updates that preserve

$H_S$3

First, $H_S$4 is updated by a pseudo-marginal Metropolis–Hastings step using an unbiased, positive likelihood estimator $H_S$5 constructed with second-order Taylor control variates. Second, conditional on the current $H_S$6, an HMC update is performed using

$H_S$7

The leapfrog proposal and the Metropolis correction both depend on the same subsample $H_S$8. This is the defining energy-conserving property.

Theoretical guarantees in the perturbed version show that the total-variation distance between the perturbed posterior $H_S$9 and the true posterior is $\theta \in \mathbb{R}^d$0, and with $\theta \in \mathbb{R}^d$1 the bias vanishes as $\theta \in \mathbb{R}^d$2. Empirically, on high-dimensional logistic regressions with $\theta \in \mathbb{R}^d$3 and $\theta \in \mathbb{R}^d$4, HMC-ECS attains acceptance rates $\theta \in \mathbb{R}^d$5–$\theta \in \mathbb{R}^d$6 and inefficiency factors $\theta \in \mathbb{R}^d$7, while using only $\theta \in \mathbb{R}^d$8–$\theta \in \mathbb{R}^d$9 of the data per iteration. Relative to standard HMC it is reported as hundreds of times faster in raw gradient-evaluation cost, and relative to SGLD and SG-HMC it is reported as $p \in \mathbb{R}^d$0–$p \in \mathbb{R}^d$1 faster and more accurate. A recurring conclusion is that subsampling itself is not the decisive ingredient; matching the subsample across trajectory generation and acceptance is.

3. Temporal-domain subsampling and perceptual filtering in video decoding

In video coding, energy conserving subsampling takes the form of temporal downscaling and perceptually motivated spatiotemporal filtering before compression. “Video Decoding Energy Reduction Using Temporal-Domain Filtering” studies discrete luminance video $p \in \mathbb{R}^d$2, its 3D Fourier transform $p \in \mathbb{R}^d$3, and a contrast-domain representation

$p \in \mathbb{R}^d$4

A spatiotemporal contrast sensitivity function (STCSF) is then used to remove frequencies deemed perceptually invisible (Herglotz et al., 2023).

The combined spatiotemporal frequency is defined as

$p \in \mathbb{R}^d$5

with

$p \in \mathbb{R}^d$6

where $p \in \mathbb{R}^d$7 cpd/pixel. The fitted STCSF is

$p \in \mathbb{R}^d$8

with $p \in \mathbb{R}^d$9, $H(\theta,p)=U(\theta)+K(p),$0, $H(\theta,p)=U(\theta)+K(p),$1, $H(\theta,p)=U(\theta)+K(p),$2, and $H(\theta,p)=U(\theta)+K(p),$3. A binary pruning mask retains only coefficients above a threshold scaled by $H(\theta,p)=U(\theta)+K(p),$4:

$H(\theta,p)=U(\theta)+K(p),$5

After inverse FFT, frame-rate reduction by integer factor $H(\theta,p)=U(\theta)+K(p),$6 is performed via temporal averaging,

$H(\theta,p)=U(\theta)+K(p),$7

The experiments use 22 HD sequences from the BVI-HFR dataset, each 512 frames at 120 fps and spatially downscaled to $H(\theta,p)=U(\theta)+K(p),$8. Compression uses x265 medium preset with $H(\theta,p)=U(\theta)+K(p),$9, quality is measured by ST-GREED, and decoding energy is measured in joules per 512-frame clip using openHEVC on an Intel i5-4670 with RAPL counters.

The principal quantitative result is that halving frame rate from 120 fps to 60 fps gives a mean $U(\theta)=-\log \pi(\theta)$0 bitrate reduction and a mean $U(\theta)=-\log \pi(\theta)$1 decoding-energy saving at constant GREED quality. Quartering frame rate to 30 fps yields a mean $U(\theta)=-\log \pi(\theta)$2 bitrate reduction and a mean $U(\theta)=-\log \pi(\theta)$3 decoding-energy saving. Spatiotemporal filtering contributes up to $U(\theta)=-\log \pi(\theta)$4 additional decoding-energy reduction, but the gains are highly content dependent: across all sequences and $U(\theta)=-\log \pi(\theta)$5, about $U(\theta)=-\log \pi(\theta)$6 of cases show positive bitrate savings and $U(\theta)=-\log \pi(\theta)$7 show decoding-energy savings, while the mean savings over all content and $U(\theta)=-\log \pi(\theta)$8 are below $U(\theta)=-\log \pi(\theta)$9.

The content dependence is central. The study reports that frame-rate reduction is effective when temporal activity is low or moderate, and that aggressive downsampling should be avoided for high structured motion. It further recommends a small STCSF scaling factor, $K(p)=\tfrac12 p^T M^{-1}p$0, together with fast content analysis and a light-weight GREED predictor for joint selection of $K(p)=\tfrac12 p^T M^{-1}p$1. Future directions explicitly include local block-based STCSF masking and machine-learning-based per-shot selection of $K(p)=\tfrac12 p^T M^{-1}p$2 under a QoE constraint.

4. Joint adaptation of resolution, chroma, and shading rate

A broader systems interpretation of energy conserving subsampling is the adaptive selection of representation granularity. “Adaptive Resolution and Chroma Subsampling for Energy-Efficient Video Coding” formalizes this as optimization over spatial resolution $K(p)=\tfrac12 p^T M^{-1}p$3, chroma format $K(p)=\tfrac12 p^T M^{-1}p$4, and target bitrate $K(p)=\tfrac12 p^T M^{-1}p$5 (Premkumar et al., 5 Feb 2026). For each candidate $K(p)=\tfrac12 p^T M^{-1}p$6, perceptual quality is $K(p)=\tfrac12 p^T M^{-1}p$7 and complexity is $K(p)=\tfrac12 p^T M^{-1}p$8, taken as average decoding time per frame $K(p)=\tfrac12 p^T M^{-1}p$9. The paper uses a normalized objective

$H$0

and chooses

$H$1

Monotonicity constraints enforce nondecreasing resolution with bitrate and nondecreasing chroma fidelity within a fixed-resolution segment.

The ARCS framework uses per-bitrate brute-force search followed by monotonicity enforcement. It is evaluated on 15 UHD (2160 p, YUV444) sequences from the SJTU 4K dataset, encoded with x265 v4.1 at the “slower” preset and decoded by the HM HEVC reference software. Quality is measured with ColorVideoVDP in JOD units; complexity is measured as average wall-clock decoding time per frame over 3 runs. Against a fixed 2160 p YUV444 baseline, ARCS at $H$2 achieves BD-rate savings of $H$3 and BD-decoding-time reduction of $H$4. At $H$5, it achieves approximately $H$6 BD-rate and $H$7 BD-time. Compared to resolution-only adaptation, ARCS adds about $H$8 percentage points of decoding-time savings at the same quality. The paper treats decoding time as a proxy for decoding energy and notes that hardware-decoder energy profiling can replace this proxy.

In graphics, “Dynamic Sampling Rate: Harnessing Frame Coherence in Graphics Applications for Energy-Efficient GPUs” uses a related principle at fragment granularity (Anglada et al., 2022). DSR operates on tiles after rasterization but before fragment shading, assigning each tile $H$9 a sampling rate $n$0. It computes a 2D DCT of tile luminance or color,

$n$1

and forms a high-frequency energy

$n$2

Thresholds $n$3 determine the coarsest admissible next-frame sampling rate, and temporal hysteresis updates $n$4 gradually to avoid flicker.

The hardware overhead is reported as small, with less than a few KB of SRAM and a few hundred gates for the DCT core. On a cycle-accurate tile-based rendering GPU model calibrated to a state-of-the-art mobile GPU, and across 12 workloads at 1080p and 60 fps target, DSR yields average speedups of $n$5 and average energy savings of $n$6. The average image quality is PSNR $n$7 dB and SSIM $n$8, and PSNR $n$9 dB and SSIM $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$0 are never violated. The method therefore exemplifies energy conserving subsampling as adaptive suppression of redundant shading work under explicit image-quality constraints.

5. Sensor-network and embedded-vision formulations

In sensor networks, the subsampling variable is the active sensor set rather than frames or fragments. “Increasing Energy Efficiency in Sensor Networks: Blue Noise Sampling and Non-Convex Matrix Completion” assumes sensors on a uniform $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$1 rectangular grid and models the field at each time as a matrix $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$2 (Majumdar et al., 2019). Empirical evidence in the paper shows that under low, medium, and high spatial correlations, only $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$3 singular values capture more than $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$4 of the energy, motivating a low-rank reconstruction model.

If only a fraction $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$5 of the $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$6 sensors is active at each time, the total network energy over $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$7 instants is

$U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$8

instead of $U(\theta)=-\sum_{k=1}^n \ell_k(\theta)$9. Blue-noise sampling is implemented by a repulsive active-set construction such as farthest-point sampling, and reconstruction is cast either as nuclear-norm minimization or as non-convex factorized matrix completion,

$H_S$00

The reported simulations use $H_S$01 fields, three correlation levels, Gaussian noise with $H_S$02, $H_S$03, and $H_S$04 standard deviation, and 1,000 trials. At $H_S$05 and zero noise in the high-correlation case, Random + convex gives RMSE $H_S$06, Quasi-crystal + convex gives RMSE $H_S$07, and Quasi-crystal + non-convex gives RMSE $H_S$08. Activating only $H_S$09 of nodes reduces acquisition plus communication energy to $H_S$10, corresponding to a $H_S$11 lifetime increase.

In embedded vision, the same idea appears as programmable ROI readout. “Adaptive Subsampling for ROI-based Visual Tracking: Algorithms and FPGA Implementation” assumes a rolling-shutter sensor with a rectangular mask $H_S$12 and subsampled frame

$H_S$13

The objective is to minimize average power while preserving AUC and minimum FPS (Iqbal et al., 2021). The system combines a heavy detector on keyframes with a Kalman-filter ROI predictor between keyframes. With state-transition matrix $H_S$14, process covariance $H_S$15, observation matrix $H_S$16, and observation covariance $H_S$17, the Kalman predictor updates the ROI through the standard prediction and correction equations and uses the predicted box to define the next sensor mask.

The sensor power model is

$H_S$18

with optimal clock

$H_S$19

so that $H_S$20 grows as $H_S$21. If the predicted box area is $H_S$22, the instantaneous power is approximated by

$H_S$23

because idle power is negligible relative to active readout. Evaluated on OTB100 and LaSOT, the ECO+KF configuration achieves AUC $H_S$24 and $H_S$25, respectively, with approximately $H_S$26 W average power and $H_S$27 FPS algorithmic performance. The YOLO+KF configuration requires approximately $H_S$28 W and reaches $H_S$29 FPS at system level. The paper identifies the keyframe interval $H_S$30 as the primary control parameter: larger $H_S$31 lowers power by reducing full-frame reads, but increases Kalman drift.

6. Recurring design principles, limitations, and open directions

Across these domains, the repeated design rule is that subsampling must be synchronized with a fidelity mechanism. In HMC-ECS, the mechanism is exact matching between the subsample used in leapfrog integration and the subsample used in the Metropolis correction. In temporal video downscaling, it is a perceptual filter based on the spatiotemporal contrast sensitivity function and a quality measure designed for temporally downscaled content. In ARCS, it is a quality–complexity objective with monotonic bitrate-ladder constraints. In DSR, it is tilewise frequency analysis plus hysteresis. In sensor networks, it is low-rank reconstruction. In ROI tracking, it is motion prediction plus periodic detector correction (Dang et al., 2017, Herglotz et al., 2023, Premkumar et al., 5 Feb 2026, Anglada et al., 2022, Majumdar et al., 2019, Iqbal et al., 2021).

A common misconception is that any reduction in sampling density automatically preserves system behavior. The literature rejects this. Naive HMC subsampling causes energy mismatch and very low acceptance. Frame-rate reduction can degrade quality heavily for high-frequency motion such as Bobblehead. Spatiotemporal filtering can worsen bitrate or QoE on some sequences. Very high $H_S$32 in ARCS can drive the ladder toward YUV420 and small resolution, which may be perceptually suboptimal in richly colored scenes. In ROI tracking, increasing the keyframe interval eventually causes predictor drift. In sensor networks, the advantage of blue-noise and non-convex recovery is empirical rather than a replacement for the theoretical matrix-completion assumptions built around purely random sampling and convex estimation.

Another recurring issue is measurement. Some studies measure energy directly: the temporal-domain video paper uses RAPL counters and reports joules per clip. Others use proxies: ARCS uses decoding time, and the paper explicitly notes that hardware energy profiling could replace $H_S$33. This suggests that “energy-conserving” is not uniform across the literature; sometimes it denotes conservation of an algorithmic Hamiltonian, and sometimes it denotes reduction of a hardware energy budget under a quality constraint.

The open directions reported in the surveyed works are similarly aligned. For video coding, they include local block-based STCSF masking and machine-learning-based selection of $H_S$34. For ARCS, they include replacing time proxies with hardware energy profiling, predicting optimal $H_S$35 from light content features, and integrating chroma adaptation into per-segment live ABR controllers such as MPEG-DASH. For ROI tracking, they include richer mask geometries, Extended KF or particle-filter motion models, multi-object tracking, and custom ASIC sensor-controller designs. A plausible implication is that future work will treat subsampling not as a fixed-rate reduction, but as a learned, content-adaptive control problem constrained simultaneously by quality, latency, and true hardware energy.