Top-k Energy Gain: Theory & Applications

• Top-k energy gain is a concept that selects the k most relevant candidates, optimizing energy metrics in varied domains such as wireless powered communication, transformer hardware, and IIoT sensing.
• In wireless networks, order statistics formalize the k-th best device selection by harvested energy or channel quality, revealing performance ceilings due to non-linear energy harvesting effects.
• Transformer and IIoT systems use top-k selection to reduce computation and communication costs, achieving significant energy efficiency gains with minimal accuracy degradation.

“Top-k energy gain” is best understood as a family of domain-specific phenomena rather than a single standardized construct. In recent arXiv literature, top-$k$ selection is used to improve energy-related objectives in at least three distinct technical settings: wireless powered communication, where devices are scheduled according to the $k$-th best channel- or energy-related statistic; transformer inference hardware, where only the $k$ largest activations are forwarded to the softmax path; and Industrial IoT sensing, where only nodes likely to belong to the top-$k$ readings are awakened for transmission. Related but semantically different uses of “energy” also arise in thermodynamic free-energy optimization and in adversarial learning, where “energy” denotes perturbation norm rather than physical power or harvested energy (Dimitropoulou et al., 2021, Dong et al., 2024, Shiraishi et al., 2024, Kolchinsky et al., 2017, Zhang et al., 2019).

1. Formal scope of top-$k$ energy gain

A precise formalization of top-$k$ selection appears in the wireless-powered setting. For $M$ i.i.d. random variables $\gamma_i$ representing a device-level metric, the order statistics are written as

$$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$

and the $k$-th best device is the one associated with $k$0. The corresponding PDF is

$k$1

This formulation makes top-$k$2 selection an order-statistical operation on a ranked population rather than a heuristic choice (Dimitropoulou et al., 2021).

Across the cited literature, the ranked object and the meaning of “energy” vary substantially.

Setting	Top-$k$3 object	Energy notion
Wireless powered communication	$k$4-th best device by e2e SNR, harvested energy, uplink gain, or worst-link	harvested RF energy and outage behavior
Transformer IMC inference	$k$5 largest activations sent to the softmax calculation block	macro energy and system-level EE in TOPS/W
IIoT sensing retrieval	nodes likely to hold the top-$k$6 readings at the deadline	total communication energy per episode
Ordered Top-$k$7 adversarial attacks	specified ordered target labels	perturbation energy $k$8
Free-energy optimization	no top-$k$9 ranking in the formulation	thermodynamic free energy gain

This diversity matters because claims about “energy gain” are not directly transferable across subfields. In one case the gain is reduced hardware energy; in another it is improved outage under non-linear energy harvesting; in another it is a lower perturbation norm or larger extractable work.

2. Order statistics and harvested-energy selection in wireless powered networks

In a wireless powered communication network with $k$0 batteryless devices, each device harvests RF energy from a dedicated energy transmitter and communicates with a common information receiver. Under the non-linear EH model, the harvested energy at device $k$1 is

$k$2

where $k$3 is the harvesting-phase duration, $k$4 is transmit power, $k$5 is the downlink channel gain, and $k$6 are rectifier parameters. The received SNR at the information receiver is

$k$7

with $k$8 the uplink channel gain and $k$9 the communication-phase duration (Dimitropoulou et al., 2021).

Four generalized selection schemes are analyzed. In SNR-based selection, the scheduled device has the $k$0-th highest e2e SNR $k$1; its outage probability is

$k$2

where $k$3 is the regularized incomplete beta function. In energy-based selection, the device with the $k$4-th most harvested energy $k$5 is chosen; the outage probability is given in closed form through an expression involving $k$6 and the modified Bessel function $k$7. Information-based selection orders devices by the uplink gain $k$8, and max-min selection ranks devices through the worst-link statistic $k$9 (Dimitropoulou et al., 2021).

The outage event is defined through the achievable-rate threshold $k$0: $k$1 Because the EH model is non-linear, increasing $k$2 does not yield unbounded improvement. The harvested energy saturates, and the high-SNR outage converges to the floor

$k$3

The paper explicitly contrasts this with a linear EH benchmark, where the outage probability decays to zero as $k$4 (Dimitropoulou et al., 2021).

The asymptotic regime $k$5 is treated with extreme value theory. For the relevant high-order statistics, the limiting law is of Gumbel type,

$k$6

and for the $k$7-th best device,

$k$8

With normalization constants $k$9 defined through

$k$0

the asymptotic outage satisfies

$k$1

A central implication is that top-$k$2 scheduling preserves tractable order-statistical structure even in massive-device regimes, but the non-linear rectifier imposes a hard performance ceiling.

3. Top-$k$3 activation selection as circuit-level energy reduction in transformers

In transformer acceleration, top-$k$4 energy gain is realized by truncating softmax computation to only the $k$5 largest activations. The Topkima approach combines top-$k$6 winner-take-all selection directly in the in-memory ADC, so only the $k$7 largest activations are sent to the softmax calculation block. This removes both the cost of full softmax evaluation and the need for a separate digital sort. A decreasing-ramp ADC detects larger MAC voltages earlier and enables early stopping once the top-$k$8 values are found (Dong et al., 2024).

For the softmax macro, the latency models are

$k$9

$M$0

and

$M$1

with

$M$2

The paper states that energy is reduced proportionally with latency and the number of computations, because reduced ADC time, reduced softmax computation, and reduced data movement all contribute directly (Dong et al., 2024).

The macro-level and system-level results are quantitatively explicit.

Accelerator	EE (TOPS/W)
ELSA	1.14
ReTransformer	0.47
TranCIM	5.10
X-Former	13.44
HARDSEA	3.73
Topkima-Former	16.84

For BERT-base on SQuAD with a $M$3 size per head of $M$4 and top-5 selection out of 384 values, Topkima-SM is approximately $M$5 and $M$6 faster than conventional and digital-top-$M$7 softmax macros, respectively, and consumes approximately $M$8 and $M$9 less energy than Conv-SM and Dtopk-SM, respectively. At system level, Topkima-Former achieves 6.70 TOPS throughput and 16.84 TOPS/W EE at 200 MHz, together with $\gamma_i$0–$\gamma_i$1 speedup and $\gamma_i$2–$\gamma_i$3 higher energy efficiency over prior IMC accelerators (Dong et al., 2024).

The accuracy trade-off is controlled by $\gamma_i$4. With a modified training scheme using top-$\gamma_i$5 only in the forward pass, $\gamma_i$6 yields only a $\gamma_i$7 to $\gamma_i$8 reduction in accuracy across ViT, distilBERT, and BERT-base on CIFAR-10, CIFAR-100, and SQuAD. The paper further reports that for $\gamma_i$9 and input length 384, the early-stop factor is $$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$0. The data therefore show that top-$$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$1 truncation is most effective when the dominant nonlinearity is dense and when a small retained set preserves model accuracy (Dong et al., 2024).

4. Content-based wake-up and top-$$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$2 sensing retrieval in IIoT

A different instantiation appears in content-based wake-up for top-$$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$3 sensing data retrieval. Here the sink must collect the most recent top-$$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$4 sensor readings before a deadline. The sink broadcasts a wake-up signal that embeds a threshold $$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$5 at time $$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$6, $$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$7 slots before the deadline. Each sensor compares its observation $$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$8 against $$\gamma_{1} \leq \gamma_{2} \leq \ldots \leq \gamma_{M},$$9; only nodes with $k$0 activate their main radios and contend for transmission using $k$1-persistent CSMA (Shiraishi et al., 2024).

The total energy per episode over $k$2 nodes is defined as

$k$3

Under content-based wake-up, the number of awakened nodes $k$4 is random, and the expected energy is

$k$5

For round-robin scheduling,

$k$6

The top-$k$7 Query Age of Information is introduced as

$k$8

and its expected version is

$k$9

This metric couples timeliness and completeness for the truly relevant top-$k$00 nodes at the deadline (Shiraishi et al., 2024).

The paper’s numerical results show that CoWu can collect top-$k$01 data with higher energy efficiency while reducing k-QAoI relative to round-robin scheduling, especially when the number of nodes is large and $k$02 is small. A specific example is given for $k$03, $k$04, and $k$05: CoWu achieves approximately 2.3 mJ versus 17.6 mJ for round-robin while obtaining lower k-QAoI. The summary further states that beyond about $k$06, CoWu is always more energy efficient for small $k$07, and that for $k$08 it can outperform round-robin up to $k$09 equal to 10 or more, corresponding to a $k$10 ratio up to 0.2 (Shiraishi et al., 2024).

The core trade-off is parametric. If $k$11 is too high, fewer than $k$12 nodes may wake, and some top-$k$13 readings may be missed; if $k$14 is too low, excess wake-ups increase contention and energy use. Likewise, large $k$15 improves communication opportunity at the cost of staler data, whereas small $k$16 improves freshness but increases deadline-miss risk. In this setting, top-$k$17 energy gain is inseparable from a freshness constraint.

5. Alternative meanings of energy gain: free energy and perturbation energy

The phrase “energy gain” also appears in two conceptually different literatures. In thermodynamic optimization, the object of study is free energy gain rather than ranked top-$k$18 selection. The central problem is to choose the initial state of a classical or quantum system so as to maximize the increase in free energy produced by a fixed process. A necessary and sufficient condition for free-energy increase is that the process be non-Gibbs-preserving, or, if Gibbs-preserving, that the final temperature $k$19 exceed the initial temperature $k$20 (Kolchinsky et al., 2017).

The comparison between optimal and suboptimal initialization is given by

$k$21

and in the generalized constrained setting by

$k$22

The optimization landscape is convex, and the same structure extends to quantum channels with quantum Kullback-Leibler divergences and von Neumann entropy. Although no top-$k$23 ranking is present here, the paper is important for separating physical free-energy gain from the other energy notions that appear in top-$k$24 work (Kolchinsky et al., 2017).

In adversarial learning, “energy” refers instead to perturbation norm. Ordered Top-$k$25 attacks require the top-$k$26 predicted labels of an adversarial example to match a specified ordered target sequence $k$27. The optimization minimizes

$k$28

where

$k$29

The target distribution $k$30 is constructed so that the target logits satisfy

$k$31

enforcing the order of the adversarial top-$k$32 predictions (Zhang et al., 2019).

In ImageNet-1000 experiments on ResNet-50, the Top-5 Average Case with 1000 iterations yields ASR $k$33 for both the modified C&W baseline and adversarial distillation, but the $k$34 norm decreases from 2.16 to 1.35 under distillation. In this literature, a lower perturbation norm is the relevant “energy gain.” The term is therefore not about hardware efficiency or communication power, but about stealthiness under a norm penalty (Zhang et al., 2019).

6. Cross-domain regularities, limitations, and common misconceptions

A recurring structural pattern is sparse retention of only the most relevant candidates. In the wireless-powered system, ranking is explicit through right-tail order statistics; in Topkima, only the $k$35 largest activations are forwarded to the nonlinear softmax block; in CoWu, a threshold is chosen so that only a small subset of nodes wakes and contends for transmission (Dimitropoulou et al., 2021, Dong et al., 2024, Shiraishi et al., 2024). This suggests that top-$k$36 energy gain is strongest when the cost of dense participation scales with population size, but the utility of the retained set is concentrated in a small number of dominant items.

A common misconception is that top-$k$37 selection always improves energy-related performance without bound. The wireless-powered analysis shows the opposite: under non-linear EH, the outage probability converges to an error floor, and the floor persists in the large-$k$38 regime unless $k$39 increases or the non-linearity parameters change (Dimitropoulou et al., 2021). In hardware, reducing $k$40 further can increase efficiency, but smaller $k$41 also increases the risk of accuracy degradation; the reported “good trade-off” is specifically tied to top-5 in the evaluated transformer models (Dong et al., 2024). In IIoT retrieval, overly aggressive thresholding can save energy while missing some true top-$k$42 data, so the gain depends on proper tuning of $k$43 and $k$44 against k-QAoI and contention (Shiraishi et al., 2024).

Another misconception is that “energy” has a single semantics across top-$k$45 research. The cited work uses at least four non-equivalent meanings: harvested RF energy and outage behavior, hardware energy and TOPS/W, communication energy per query episode, and perturbation norm; thermodynamic free energy adds yet another meaning (Dong et al., 2024, Shiraishi et al., 2024, Zhang et al., 2019, Kolchinsky et al., 2017). Consequently, cross-paper comparisons are meaningful only within a shared operational definition.

A plausible cross-domain implication is that top-$k$46 methods are most effective when they simultaneously exploit heavy-tailed relevance and avoid expensive dense computation or communication. The existing literature also shows that their practical value is bounded by non-idealities: rectifier saturation, accuracy loss under aggressive truncation, contention under imperfect thresholding, or the fact that some “energy gains” are informational rather than physical.