A theoretical study says a streaming solver for corrupted linear equations can converge with corruption levels up to approximately 7% in the paper's Massart model. In a separate analysis of oblivious noise, the reported peak is approximately 32%. Those figures come from an arXiv preprint and describe mathematical guarantees under the model it analyzes, not universal limits for every noisy system.
The work is an arXiv preprint, version 1, dated 28 August 2026. It focuses on how much corruption a streaming version of Quantile Randomized Kaczmarz, or QRK, can tolerate while keeping its error moving toward the true solution, and how many fresh measurements each step requires.
A solver built for the stream
QRK is built for a streaming setting, where the algorithm receives fresh measurements as it works. At each iteration, it uses D fresh measurements to calculate an empirical q-quantile, then takes one additional fresh measurement for the update. A quantile acts as a cutoff chosen from a set of values. The total at each step is the quantile subsample plus one more fresh measurement.
The fresh-sample design is central to the theory. The measurement vectors in the model are independent and identically distributed, or i.i.d., and uniform over the unit sphere. Because each iteration draws new data, the guarantees apply to the stated streaming setup, which is why the reported percentages should be read as assumptions-based bounds.
Two models, different ceilings
The paper distinguishes Massart and oblivious corruption according to whether the corruption values may be chosen adversarially. Under the Massart analysis, it states a linear-convergence guarantee for corruption levels up to about 7%. Its numerical calculation puts the theoretical peak at approximately 0.069 when q, the quantile parameter, is 0.85. Put simply, the claim is that the solver's error falls at a predictable rate as it repeats the update.
The oblivious calculation gives a much higher theoretical peak, approximately 0.320 at q = 0.65. It is not a single head-to-head score with the Massart figure, because the noise model and the quantile setting differ. The paper also reports a bound of 0.397 at q = 0.60 when the corruption is known to be Gaussian, above the 0.320 oblivious value. The comparison underscores how dependent the stated threshold is on the corruption model and its assumptions.
How many fresh measurements are enough?
The analysis turns those theoretical conditions into concrete sampling requirements. With q = 0.75, a corruption level of 0.01 and a run of 20,000 iterations, the reported Massart calculation guarantees convergence when D, the quantile-subsample size, is 25 at each iteration. Under oblivious noise with the same settings, the reported requirement is D = 13. The Massart bound is described as conservative, while both figures are specific to the chosen parameters and model.
Beyond these examples, the conclusion gives a scaling rule: the number of fresh quantile samples needed per iteration grows on the order of the logarithm of the total iteration horizon. A longer run therefore calls for more fresh samples, but the required batch need not grow in direct proportion to the number of iterations within the analyzed assumptions.
The simulations match the broad pattern
Simulations were used to see whether the theoretical thresholds lined up with empirical success transitions. The tests fixed the problem dimension at 100, set q to 0.8, used a failure tolerance of 0.1 and a success-criterion parameter of 0.05. For every parameter pair, the authors ran 100 independent trials, starting the algorithm at x0 = 0.
Across both corruption models, the theoretical subsample thresholds tracked the empirical transitions, although they became more conservative as corruption increased. Because the simulations used author-selected criteria and parameter grids, the comparison is qualitative.
The bounds stay inside the model
The conclusion says that logarithmic per-iteration subsamples suffice under the analyzed models. But the guarantees remain assumption-dependent. They rely on i.i.d. measurement vectors uniformly distributed on the unit sphere, and the numerical constants use the paper's normal approximations. The reported Massart subsample bound is also described as conservative.
Taken together, the work offers mathematical benchmarks inside the analyzed streaming models. The 7% Massart result and 32% oblivious peak should not be read as universal corruption limits, because each figure belongs to a particular noise model, quantile setting and set of assumptions.
Paper data and sources
Original title: Quantile Randomized Kaczmarz for Streaming Linear Systems with Massart Noise
Authors: Emeric Battaglia, Jian-Feng Cai, Junren Chen et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
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