Recovering a Missing Sensor Channel Without Retraining the Model

Change the inputs, not the model

In their November 2026 preprint, Björn Ludwig, Alexander Windmann, René Heesch and Oliver Niggemann investigate a specific question: can a time-series reconstructor become an imputer—a system for estimating missing values—without retraining its weights?

Their method, called test-time optimization, starts with an autoencoder trained to reconstruct time-series windows. Training uses available measurements, fills native gaps with a baseline value and scores reconstruction only where measurements exist. Crucially, the reconstructors receive no artificial masking or full-channel dropout during training.

At inference, the model’s weights stay frozen. Missing inputs start at their channel’s training-set mean, while observed inputs remain fixed. The algorithm repeatedly adjusts only the missing inputs, backpropagating through the network to reduce reconstruction error on observed coordinates. Hidden reference values are not used in this optimization.

The final estimates come from the network’s output, not directly from the optimized input variables. Those predictions fill missing positions; all available measurements remain unchanged.

Why matching known readings can help

The approach depends on learned relationships between channels. If changing a candidate missing trajectory changes the network’s reconstruction of observed measurements, those observed reconstruction errors can guide the search.

The authors describe this dependence with a cross-Jacobian: a mathematical description of how missing inputs influence observed outputs. A model that simply copies each observed input independently offers no useful recovery signal. Its observed reconstruction can be correct regardless of what fills the missing channel.

The paper’s theoretical result is local and conditional. Under differentiability and exact-self-consistency assumptions, sufficient cross-Jacobian rank isolates a nearby self-consistent solution. It does not guarantee that the solution matches the missing measurements or that distant alternatives do not exist.

An optional temporal-smoothness penalty discourages abrupt changes between adjacent missing values. This can regularize weakly constrained solutions, but cannot eliminate their underlying ambiguity.

What the experiments actually test

The main evaluation uses five real-world datasets: BeijingAir and ItalyAir for air quality, ETTh1 for energy and temperature, Electricity for electricity load, and PeMS for traffic. Their respective window lengths are 24, 12, 48, 96 and 24 time steps. Data are split chronologically and standardized using training-split statistics.

Exactly one complete channel is synthetically hidden in each test window, with channel selection assigned cyclically. Errors are scored only on hidden entries that originally had measurements. BeijingAir also has 1.6% native missingness; the other datasets have none.

The researchers test LSTM, MTAD-GAT, TCN-AE and TransformerAE reconstructors, reporting means and standard deviations across six random seeds. Optimization uses Adam with a learning rate of 0.01, a maximum of 5,000 iterations and objective-based early stopping—not selection of the iteration with the lowest hidden-value error.

Consistent improvement is not universal superiority

Without the smoothness penalty, optimization reduces mean absolute error relative to the same frozen model’s one-pass reconstruction in 19 of 20 model–dataset combinations. The authors report a 12.9% average relative reduction across those pairs.

Benefits vary sharply. Averaging relative improvements across architectures gives 29.1% on ETTh1, but only 0.4% on Electricity and 0.5% on PeMS. The exception is TCN-AE on Electricity, where refinement increases error by approximately 0.8%.

The optional prior is also mixed: it improves 11 combinations, leaves two unchanged at reported precision and worsens seven.

Against specialized mask-aware imputers, the best refined reconstructor has the lowest reported mean error on ETTh1 and Electricity, but trails on BeijingAir, ItalyAir and PeMS. On ETTh1, for example, LSTM refinement with the prior achieves 0.2898 ± 0.0416, compared with 0.3180 ± 0.0111 for the strongest reported mask-aware result.

These comparisons use post-hoc per-dataset minima across multiple configurations, not one model selected in advance. Baselines receive either structured artificial masks during training or matched full-channel dropout; reconstructors do not. On Electricity, the unrefined TCN-AE actually outperforms every refined reconstructor, despite the refined family beating the external baselines.

The computational and practical limits

Accuracy gains require repeated forward and backward passes. Comparing the fastest measured configuration from each family on identical hardware, the authors report optimization as 6.7–174.7 times slower than one-shot inference. Those fastest optimization configurations typically stop around 200 iterations.

The evidence supports a conditional adapter, not a replacement for dedicated imputers. The study does not establish performance for multiple missing channels, arbitrary reconstructors or deployed-model reuse. It produces point estimates rather than uncertainty distributions, and does not test downstream forecasting, anomaly detection or maintenance savings. The authors recommend flagging imputed values rather than treating them as measured ground truth.