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This work presents the development of a protocol for the measurement of collisional deexcitation rate constants of N$_2$(C$^3\Pi_u$ , $\nu$ = 0-4) States by O$_2$, N$_2$, H$_2$, CO$_2$ and CH$_4$. Excited nitrogen is produced using nanosecond repetitively pulsed discharges. The applied voltage and the repetition frequencies are adjusted to minimize rotational and gas heating while keeping high electronic and vibrational excitation levels. The populations of the vibrational levels are measured using optical emission spectroscopy of the N$_2$(C $\to$ B,$\Delta_\nu=0$) vibrational bands. Rate coefficients for each vibrational level are then determined from the measured lifetimes. The results are compared to the available literature and new values are provided for the missing vibrational levels., Comment: AIAA SCITECH 2025 Forum, Jan 2025, Orlando, France.

The explainability of machine learning algorithms is crucial, and numerous methods have emerged recently. Local, post-hoc methods assign an attribution score to each feature, indicating its importance for the prediction. However, these methods require recalculating explanations for each example. On the other side, while there exist global approaches they often produce explanations that are either overly simplistic and unreliable or excessively complex. To bridge this gap, we propose GLEAMS, a novel method that partitions the input space and learns an interpretable model within each sub-region, thereby providing both faithful local and global surrogates. We demonstrate GLEAMS' effectiveness on both synthetic and real-world data, highlighting its desirable properties and human-understandable insights.

To leverage the angular resolution of interferometry at high contrast, one must employ specialized beam-combiners called interferometric nullers. Nullers discard part of the astrophysical information to optimize the recording of light present in the dark fringe of the central source. Asgard/NOTT will deploy a beam-combination scheme offering good instrumental noise rejection when phased appropriately, but for which information is degenerate on the outputs, prompting a dedicated tuning strategy using the science detector. The dispersive effect of water vapor can be corrected with prisms forming a variable thickness of glass. But observations in the L band suffer from an additional and important chromatic effect due to longitudinal atmospheric dispersion coming from a resonance of CO2 at 4.3 micron. To compensate for this effect efficiently, a novel type of compensation device will be deployed leveraging a gas cell of variable length at ambient pressure. After reviewing the impact of water vapor and CO2, we present the design of this atmospheric dispersion compensation device and describe a strategy to maintain this tuning on-sky., Comment: 7 pages, 5 figures, Proc of SPIE Astronomical Telescopes + Instrumentation 2024 Yokohama, Japan, Optical and Infrared Interferometry and Imaging VI

Smectic liquid crystals can be viewed as model systems for lamellar structures for which there has been extensive theoretical development. We demonstrate that a nonlinear energy description is required with respect to the usual Landau-de Gennes elasticity in order to explain the observed layer spacing of highly curved smectic layers. Using X-ray diffraction we have quantitatively determined the dilation of bent layers distorted by antagonistic anchoring (as high as 1.8% of dilation for the most bent smectic layers) and accurately described it by the minimal nonlinear expression for energy. We observe a 1{\deg} tilt of planar layers that are connected to the curved layers. This value is consistent with simple energetic calculations, demonstrating how the bending energy impacts the overall structure of a thin distorted smectic film. Finally, we show that combined X-ray measurements and theoretical modeling allow for the quantitative determination of the number of curved smectic layers and of the resulting thickness of the dilated region with unprecedented precision., Comment: 6 pages with 6 figures

Nulling interferometry is a powerful observing technique to reach exoplanets and circumstellar dust at separations too small for direct imaging with single-dish telescopes and too large for indirect methods. With near-future instrumentation, it bears the potential to detect young, hot planets near the snow lines of their host stars. A future space mission could detect and characterize a large number of rocky, habitable-zone planets around nearby stars at thermal-infrared wavelengths. The null self-calibration is a method aiming at modelling the statistical distribution of the nulled signal. It has proven to be more sensitive and accurate than average-based data reduction methods in nulling interferometry. This statistical approach opens the possibility of designing a GPU-based Python package to reduce the data from any of these instruments, by simply providing the data and a simulator of the instrument. GRIP is a toolbox to reduce nulling and interferometric data based on the statistical self-calibration method. In this article, we present the main features of GRIP as well as applications on real data.

Asgard/NOTT is an ERC-funded project hosted at KU Leuven and is part of a new visitor instrumental suite, called Asgard, under preparation for the Very Large Telescope Interferometer (VLTI). Leveraging nulling capabilities and the long VLTI baselines, it is optimized for high-contrast imaging of the snow line region around young nearby main-sequence stars. This will enable the characterization of the atmosphere of young giant exoplanets and warm/hot exozodiacal dust with spectroscopy in the L'-band (3.5-4.0$\mu$m). In this work, we present the first lab assembly of the instrument done at KU Leuven and the technical solutions to tackle the challenge of performing nulling in the mid-infrared despite the thermal background. The opto-mechanical design of the warm optics and the injection system for the photonic chip are described. The alignment procedure used to assemble the system is also presented. Finally, the first experimental results, including fringes and null measurements, are given and confirm the adequacy of the bench to test and optimize the Asgard/NOTT instrument., Comment: 13 pages (incl. 14 figures); Proc. SPIE Astronomical Telescopes + Instrumentation 2024 (Yokohama; Japan), Optical and Infrared Interferometry and Imaging IX

ESO's Very Large Telescope Interferometer has a history of record-breaking discoveries in astrophysics and significant advances in instrumentation. The next leap forward is its new visitor instrument, called Asgard. It comprises four natively collaborating instruments: HEIMDALLR, an instrument performing both fringe tracking and stellar interferometry simultaneously with the same optics, operating in the K band; Baldr, a Strehl optimizer in the H band; BIFROST, a spectroscopic combiner to study the formation processes and properties of stellar and planetary systems in the Y-J-H bands; and NOTT, a nulling interferometer dedicated to imaging nearby young planetary systems in the L band. The suite is in its integration phase in Europe and should be shipped to Paranal in 2025. In this article, we present details of the alignment and calibration unit, the observing modes, the integration plan, the software architecture, and the roadmap to completion of the project.

NOTT (formerly Hi-5) is the L'-band (3.5-4.0~microns) nulling interferometer of Asgard, an instrument suite in preparation for the VLTI visitor focus. The primary scientific objectives of NOTT include characterizing (i) young planetary systems near the snow line, a critical region for giant planet formation, and (ii) nearby main-sequence stars close to the habitable zone, with a focus on detecting exozodiacal dust that could obscure Earth-like planets. In 2023-2024, the final warm optics have been procured and assembled in a new laboratory at KU Leuven. First fringes and null measurements were obtained using a Gallium Lanthanum Sulfide (GLS) photonic chip that was also tested at cryogenic temperatures. In this paper, we present an overall update of the NOTT project with a particular focus on the cold mechanical design, the first results in the laboratory with the final NOTT warm optics, and the ongoing Asgard integration activities. We also report on other ongoing activities such as the characterization of the photonic chip (GLS, LiNbO3, SiO), the development of the exoplanet science case, the design of the dispersion control module, and the progress with the self-calibration data reduction software., Comment: 11 pages (incl. 5 figures); Proc. SPIE Astronomical Telescopes + Instrumentation 2024 (Yokohama; Japan), Optical and Infrared Interferometry and Imaging VI

We introduce Meta 3D Gen (3DGen), a new state-of-the-art, fast pipeline for text-to-3D asset generation. 3DGen offers 3D asset creation with high prompt fidelity and high-quality 3D shapes and textures in under a minute. It supports physically-based rendering (PBR), necessary for 3D asset relighting in real-world applications. Additionally, 3DGen supports generative retexturing of previously generated (or artist-created) 3D shapes using additional textual inputs provided by the user. 3DGen integrates key technical components, Meta 3D AssetGen and Meta 3D TextureGen, that we developed for text-to-3D and text-to-texture generation, respectively. By combining their strengths, 3DGen represents 3D objects simultaneously in three ways: in view space, in volumetric space, and in UV (or texture) space. The integration of these two techniques achieves a win rate of 68% with respect to the single-stage model. We compare 3DGen to numerous industry baselines, and show that it outperforms them in terms of prompt fidelity and visual quality for complex textual prompts, while being significantly faster.

We present Meta 3D AssetGen (AssetGen), a significant advancement in text-to-3D generation which produces faithful, high-quality meshes with texture and material control. Compared to works that bake shading in the 3D object's appearance, AssetGen outputs physically-based rendering (PBR) materials, supporting realistic relighting. AssetGen generates first several views of the object with factored shaded and albedo appearance channels, and then reconstructs colours, metalness and roughness in 3D, using a deferred shading loss for efficient supervision. It also uses a sign-distance function to represent 3D shape more reliably and introduces a corresponding loss for direct shape supervision. This is implemented using fused kernels for high memory efficiency. After mesh extraction, a texture refinement transformer operating in UV space significantly improves sharpness and details. AssetGen achieves 17% improvement in Chamfer Distance and 40% in LPIPS over the best concurrent work for few-view reconstruction, and a human preference of 72% over the best industry competitors of comparable speed, including those that support PBR. Project page with generated assets: https://assetgen.github.io, Comment: Project Page: https://assetgen.github.io

We consider the Gross-Pitaevskii equation with a confining ring potential with a Gaussian profile. By introducing a rotating sinusoidal perturbation, we numerically highlight the nucleation of quantum vortices in a particular regime throughout the dynamics. Numerical computations are made via a Strang splitting time integration and a two-point flux approximation Finite Volume scheme based on a particular admissible triangulation. We also develop numerical algorithms for vortex tracking adapted to our finite volume framework.

CAM-based methods are widely-used post-hoc interpretability method that produce a saliency map to explain the decision of an image classification model. The saliency map highlights the important areas of the image relevant to the prediction. In this paper, we show that most of these methods can incorrectly attribute an important score to parts of the image that the model cannot see. We show that this phenomenon occurs both theoretically and experimentally. On the theory side, we analyze the behavior of GradCAM on a simple masked CNN model at initialization. Experimentally, we train a VGG-like model constrained to not use the lower part of the image and nevertheless observe positive scores in the unseen part of the image. This behavior is evaluated quantitatively on two new datasets. We believe that this is problematic, potentially leading to mis-interpretation of the model's behavior., Comment: Accepted for publication at ECML 2024 (28 pages, 9 figures)

Asgard/NOTT (previously Hi-5) is a European Research Council (ERC)-funded project hosted at KU Leuven and a new visitor instrument for the Very Large Telescope Interferometer (VLTI). Its primary goal is to image the snow line region around young stars using nulling interferometry in the L-band (3.5 to 4.0)$\mu$m, where the contrast between exoplanets and their host stars is advantageous. The breakthrough is the use of a photonic beam combiner, which only recently allowed the required theoretical raw contrast of $10^{-3}$ in this spectral range. Nulling interferometry observations of exoplanets also require a high degree of balancing between the four pupils of the VLTI in terms of intensity, phase, and polarization. The injection into the beam combiner and the requirements of nulling interferometry are driving the design of the warm optics and the injection system. The optical design up to the beam combiner is presented. It offers a technical solution to efficiently couple the light from the VLTI into the beam combiner. During the coupling, the objective is to limit throughput losses to 5% of the best expected efficiency for the injection. To achieve this, a list of different loss sources is considered with their respective impact on the injection efficiency. Solutions are also proposed to meet the requirements on beam balancing for intensity, phase, and polarization. The different properties of the design are listed, including the optics used, their alignment and tolerances, and their impact on the instrumental performances in terms of throughput and null depth. The performance evaluation gives an expected throughput loss of less than <6.4% of the best efficiency for the injection and a null depth of $\sim2.10^{-3}$, mainly from optical path delay errors outside the scope of this work., Comment: Accepted for publication in JATIS. 23 pages, 11 figures, 8 tables

Phase transitions are prevalent throughout physics, spanning thermal phenomena like water boiling to magnetic transitions in solids. They encompass cosmological phase transitions in the early universe and the transition into a quark-gluon plasma in high-energy collisions. Quantum phase transitions, particularly intriguing, occur at temperatures near absolute zero and are driven by quantum fluctuations rather than thermal ones. The strength of the fluctuations is very sensitive to the dimensionality of the physical systems, which determines the existence and nature of phase transitions. Low-dimensional systems often exhibit suppression of phase transitions, while high-dimensional systems tend to exhibit mean-field-like behavior. The localization-delocalization Anderson transition stands out among quantum phase transitions, as it is thought to retain its non-mean-field character across all dimensions. This work marks the first observation and characterization of the Anderson transition in four dimensions using ultracold atoms as a quantum simulator with synthetic dimensions. We characterize the universal dynamics in the vicinity of the phase transition. We measure the critical exponents describing the scale-invariant properties of the critical dynamics, which are shown to obey Wegner's scaling law. Our work is the first experimental demonstration that the Anderson transition is not mean-field in dimension four., Comment: 10 pages; 5 figures

Attention-based architectures, in particular transformers, are at the heart of a technological revolution. Interestingly, in addition to helping obtain state-of-the-art results on a wide range of applications, the attention mechanism intrinsically provides meaningful insights on the internal behavior of the model. Can these insights be used as explanations? Debate rages on. In this paper, we mathematically study a simple attention-based architecture and pinpoint the differences between post-hoc and attention-based explanations. We show that they provide quite different results, and that, despite their limitations, post-hoc methods are capable of capturing more useful insights than merely examining the attention weights., Comment: Accepted at ICML 2024

Ensemble methods combine the predictions of several base models. We study whether or not including more models always improves their average performance. This question depends on the kind of ensemble considered, as well as the predictive metric chosen. We focus on situations where all members of the ensemble are a priori expected to perform as well, which is the case of several popular methods such as random forests or deep ensembles. In this setting, we show that ensembles are getting better all the time if, and only if, the considered loss function is convex. More precisely, in that case, the average loss of the ensemble is a decreasing function of the number of models. When the loss function is nonconvex, we show a series of results that can be summarised as: ensembles of good models keep getting better, and ensembles of bad models keep getting worse. To this end, we prove a new result on the monotonicity of tail probabilities that may be of independent interest. We illustrate our results on a medical prediction problem (diagnosing melanomas using neural nets) and a "wisdom of crowds" experiment (guessing the ratings of upcoming movies).

Interpretability is essential for machine learning models to be trusted and deployed in critical domains. However, existing methods for interpreting text models are often complex, lack mathematical foundations, and their performance is not guaranteed. In this paper, we propose FRED (Faithful and Robust Explainer for textual Documents), a novel method for interpreting predictions over text. FRED offers three key insights to explain a model prediction: (1) it identifies the minimal set of words in a document whose removal has the strongest influence on the prediction, (2) it assigns an importance score to each token, reflecting its influence on the model's output, and (3) it provides counterfactual explanations by generating examples similar to the original document, but leading to a different prediction. We establish the reliability of FRED through formal definitions and theoretical analyses on interpretable classifiers. Additionally, our empirical evaluation against state-of-the-art methods demonstrates the effectiveness of FRED in providing insights into text models.

Algorithmic recourse provides explanations that help users overturn an unfavorable decision by a machine learning system. But so far very little attention has been paid to whether providing recourse is beneficial or not. We introduce an abstract learning-theoretic framework that compares the risks (i.e., expected losses) for classification with and without algorithmic recourse. This allows us to answer the question of when providing recourse is beneficial or harmful at the population level. Surprisingly, we find that there are many plausible scenarios in which providing recourse turns out to be harmful, because it pushes users to regions of higher class uncertainty and therefore leads to more mistakes. We further study whether the party deploying the classifier has an incentive to strategize in anticipation of having to provide recourse, and we find that sometimes they do, to the detriment of their users. Providing algorithmic recourse may therefore also be harmful at the systemic level. We confirm our theoretical findings in experiments on simulated and real-world data. All in all, we conclude that the current concept of algorithmic recourse is not reliably beneficial, and therefore requires rethinking., Comment: 24 pages, 8 figures, 5 tables

In many scenarios, the interpretability of machine learning models is a highly required but difficult task. To explain the individual predictions of such models, local model-agnostic approaches have been proposed. However, the process generating the explanations can be, for a user, as mysterious as the prediction to be explained. Furthermore, interpretability methods frequently lack theoretical guarantees, and their behavior on simple models is frequently unknown. While it is difficult, if not impossible, to ensure that an explainer behaves as expected on a cutting-edge model, we can at least ensure that everything works on simple, already interpretable models. In this paper, we present a theoretical analysis of Anchors (Ribeiro et al., 2018): a popular rule-based interpretability method that highlights a small set of words to explain a text classifier's decision. After formalizing its algorithm and providing useful insights, we demonstrate mathematically that Anchors produces meaningful results when used with linear text classifiers on top of a TF-IDF vectorization. We believe that our analysis framework can aid in the development of new explainability methods based on solid theoretical foundations., Comment: arXiv admin note: substantial text overlap with arXiv:2205.13789

A fundamental issue in machine learning is the robustness of the model with respect to changes in the input. In natural language processing, models typically contain a first embedding layer, transforming a sequence of tokens into vector representations. While the robustness with respect to changes of continuous inputs is well-understood, the situation is less clear when considering discrete changes, for instance replacing a word by another in an input sentence. Our work formally proves that popular embedding schemes, such as concatenation, TF-IDF, and Paragraph Vector (a.k.a. doc2vec), exhibit robustness in the H\"older or Lipschitz sense with respect to the Hamming distance. We provide quantitative bounds for these schemes and demonstrate how the constants involved are affected by the length of the document. These findings are exemplified through a series of numerical examples., Comment: Accepted to ICML 2023. 33 pages, 10 figures

The Very Large Telescope Interferometer is one of the most proficient observatories in the world for high angular resolution. Since its first observations, it has hosted several interferometric instruments operating in various bandwidths in the infrared. As a result, the VLTI has yielded countless discoveries and technological breakthroughs. Here, we introduce a new concept for the VLTI, Asgard: an instrumental suite comprised of four natively collaborating instruments: BIFROST, a combiner whose main science case is studying the formation processes and properties of stellar and planetary systems; NOTT, a nulling interferometer dedicated to imaging young nearby planetary systems in the L band; HEIMDALLR, an all-in-one instrument performing both fringe tracking and stellar interferometry with the same optics; Baldr, a Strehl optimiser. These instruments share common goals and technologies. The goals are diverse astrophysical cases such as the study of the formation and evolution processes of binary systems, exoplanetary systems and protoplanetary disks, the characterization of orbital parameters and spin-orbit alignment of multiple systems, the characterization of the exoplanets, and the study of exozodiacal disks. Thus, the idea of this suite is to make the instruments interoperable and complementary to deliver unprecedented sensitivity and accuracy from the J to M bands to meet these goals. The interoperability of the Asgard instruments and their integration in the VLTI are major challenges for this project.

A wide variety of model explanation approaches have been proposed in recent years, all guided by very different rationales and heuristics. In this paper, we take a new route and cast interpretability as a statistical inference problem. We propose a general deep probabilistic model designed to produce interpretable predictions. The model parameters can be learned via maximum likelihood, and the method can be adapted to any predictor network architecture and any type of prediction problem. Our method is a case of amortized interpretability models, where a neural network is used as a selector to allow for fast interpretation at inference time. Several popular interpretability methods are shown to be particular cases of regularised maximum likelihood for our general model. We propose new datasets with ground truth selection which allow for the evaluation of the features importance map. Using these datasets, we show experimentally that using multiple imputation provides more reasonable interpretations., Comment: 10 pages, 22 figures, published at ICLR 2023

Context: NOTT (formerly Hi-5) is a new high-contrast L' band (3.5-4.0 \textmu m) beam combiner for the VLTI with the ambitious goal to be sensitive to young giant exoplanets down to 5 mas separation around nearby stars. The performance of nulling interferometers in these wavelengths is affected both by fundamental noise from the background and by the contributions of instrumental noises. This motivates the development of end-to-end simulations to optimize these instruments. Aims: To enable the performance evaluation and inform the design of such instruments on the current and future infrastructures, taking into account the different sources of noise, and their correlation. Methods: SCIFYsim is an end-to-end simulator for single mode filtered beam combiners, with an emphasis on nulling interferometers. It is used to compute a covariance matrix of the errors. Statistical detection tests based on likelihood ratios are then used to compute compound detection limits for the instrument. Results: With the current assumptions on the performance of the wavefront correction systems, the errors are dominated by correlated instrumental errors down to stars of magnitude 6-7 in the L band, beyond which thermal background from the telescopes and relay system becomes dominant. Conclusions: SCIFYsim is suited to anticipate some of the challenges of design, tuning, operation and signal processing for integrated optics beam combiners. The detection limits found for this early version of NOTT simulation with the unit telescopes are compatible with detections at contrasts up to $10^5$ in the L band at separations of 5 to 80 mas around bright stars.

Hi-5 is the L'-band (3.5-4.0 $\mu$m) high-contrast imager of Asgard, an instrument suite in preparation for the visitor focus of the VLTI. The system is optimized for high-contrast and high-sensitivity imaging within the diffraction limit of a single UT/AT telescope. It is designed as a double-Bracewell nulling instrument producing spectrally-dispersed (R=20, 400, or 2000) complementary nulling outputs and simultaneous photometric outputs for self-calibration purposes. In this paper, we present an update of the project with a particular focus on the overall architecture, opto-mechanical design of the warm and cold optics, injection system, and development of the photonic beam combiner. The key science projects are to survey (i) nearby young planetary systems near the snow line, where most giant planets are expected to be formed, and (ii) nearby main sequence stars near the habitable zone where exozodiacal dust that may hinder the detection of Earth-like planets. We present an update of the expected instrumental performance based on full end-to-end simulations using the new GRAVITY+ specifications of the VLTI and the latest planet formation models., Comment: 16 pages, 9 figures, SPIE 2022 "Astronomical Telescopes and Instrumentation" manuscript 12183-16

Complex machine learning algorithms are used more and more often in critical tasks involving text data, leading to the development of interpretability methods. Among local methods, two families have emerged: those computing importance scores for each feature and those extracting simple logical rules. In this paper we show that using different methods can lead to unexpectedly different explanations, even when applied to simple models for which we would expect qualitative coincidence. To quantify this effect, we propose a new approach to compare explanations produced by different methods., Comment: Accepted to XAIE ICPR 2022, the 2-nd Workshop on Explainable and Ethical AI, ICPR 2022

Inspired from quantum Monte Carlo, by using unbiased estimators all the time and sampling discrete and continuous variables at the same time using Metropolis algorithm, we present a novel, fast, and accurate high performance Monte Carlo Parametric Expectation Maximization (MCPEM) algorithm. We named it Randomized Parametric Expectation Maximization (RPEM). In particular, we compared RPEM with Monolix's SAEM and Certara's QRPEM for a realistic two-compartment Voriconazole model with ordinary differential equations (ODEs) and using simulated data. We show that RPEM is 3 to 4 times faster than SAEM and QRPEM, and more accurate than them in reconstructing the population parameters., Comment: 28 pages, 6 figures, 2 tables

Anchors (Ribeiro et al., 2018) is a post-hoc, rule-based interpretability method. For text data, it proposes to explain a decision by highlighting a small set of words (an anchor) such that the model to explain has similar outputs when they are present in a document. In this paper, we present the first theoretical analysis of Anchors, considering that the search for the best anchor is exhaustive. After formalizing the algorithm for text classification, we present explicit results on different classes of models when the vectorization step is TF-IDF, and words are replaced by a fixed out-of-dictionary token when removed. Our inquiry covers models such as elementary if-then rules and linear classifiers. We then leverage this analysis to gain insights on the behavior of Anchors for any differentiable classifiers. For neural networks, we empirically show that the words corresponding to the highest partial derivatives of the model with respect to the input, reweighted by the inverse document frequencies, are selected by Anchors., Comment: Accepted to AISTATS 2023. 9+2 page paper, 21-page appendix

Anderson localization is the ubiquitous phenomenon of inhibition of transport of classical and quantum waves in a disordered medium. In dimension one, it is well known that all states are localized, implying that the distribution of an initially narrow wave-packet released in a disordered potential will, at long time, decay exponentially on the scale of the localization length. However, the exact shape of the stationary localized distribution differs from a purely exponential profile and has been computed almost fifty years ago by Gogolin. Using the atomic quantum kicked rotor, a paradigmatic quantum simulator of Anderson localization physics, we study this asymptotic distribution by two complementary approaches. First, we discuss the connection of the statistical properties of the system's localized eigenfunctions and their exponential decay with the localization length of the Gogolin distribution. Next, we make use of our experimental platform, realizing an ideal Floquet disordered system, to measure the long-time probability distribution and highlight the very good agreement with the analytical prediction compared to the purely exponential one over 3 orders of magnitude., Comment: 10 pages, 8 figures

Quickshift is a popular algorithm for image segmentation, used as a preprocessing step in many applications. Unfortunately, it is quite challenging to understand the hyperparameters' influence on the number and shape of superpixels produced by the method. In this paper, we study theoretically a slightly modified version of the quickshift algorithm, with a particular emphasis on homogeneous image patches with i.i.d. pixel noise and sharp boundaries between such patches. Leveraging this analysis, we derive a simple heuristic to scale quickshift hyperparameters with respect to the image size, which we check empirically., Comment: 33 pages, 16 figures. Accepted to AISTATS 2022