@inproceedings{13146,
  abstract     = {A recent line of work has analyzed the theoretical properties of deep neural networks via the Neural Tangent Kernel (NTK). In particular, the smallest eigenvalue of the NTK has been related to the memorization capacity, the global convergence of gradient descent algorithms and the generalization of deep nets. However, existing results either provide bounds in the two-layer setting or assume that the spectrum of the NTK matrices is bounded away from 0 for multi-layer networks. In this paper, we provide tight bounds on the smallest eigenvalue of NTK matrices for deep ReLU nets, both in the limiting case of infinite widths and for finite widths. In the finite-width setting, the network architectures we consider are fairly general: we require the existence of a wide layer with roughly order of N neurons, N being the number of data samples; and the scaling of the remaining layer widths is arbitrary (up to logarithmic factors). To obtain our results, we analyze various quantities of independent interest: we give lower bounds on the smallest singular value of hidden feature matrices, and upper bounds on the Lipschitz constant of input-output feature maps.},
  author       = {Nguyen, Quynh and Mondelli, Marco and Montufar, Guido},
  booktitle    = {Proceedings of the 38th International Conference on Machine Learning},
  isbn         = {9781713845065},
  issn         = {2640-3498},
  location     = {Virtual},
  pages        = {8119--8129},
  publisher    = {ML Research Press},
  title        = {{Tight bounds on the smallest Eigenvalue of the neural tangent kernel for deep ReLU networks}},
  volume       = {139},
  year         = {2021},
}

@inproceedings{13147,
  abstract     = {We investigate fast and communication-efficient algorithms for the classic problem of minimizing a sum of strongly convex and smooth functions that are distributed among n
 different nodes, which can communicate using a limited number of bits. Most previous communication-efficient approaches for this problem are limited to first-order optimization, and therefore have \emph{linear} dependence on the condition number in their communication complexity. We show that this dependence is not inherent: communication-efficient methods can in fact have sublinear dependence on the condition number. For this, we design and analyze the first communication-efficient distributed variants of preconditioned gradient descent for Generalized Linear Models, and for Newton’s method. Our results rely on a new technique for quantizing both the preconditioner and the descent direction at each step of the algorithms, while controlling their convergence rate. We also validate our findings experimentally, showing faster convergence and reduced communication relative to previous methods.},
  author       = {Alimisis, Foivos and Davies, Peter and Alistarh, Dan-Adrian},
  booktitle    = {Proceedings of the 38th International Conference on Machine Learning},
  isbn         = {9781713845065},
  issn         = {2640-3498},
  location     = {Virtual},
  pages        = {196--206},
  publisher    = {ML Research Press},
  title        = {{Communication-efficient distributed optimization with quantized preconditioners}},
  volume       = {139},
  year         = {2021},
}

@article{14117,
  abstract     = {The two fields of machine learning and graphical causality arose and are developed separately. However, there is, now, cross-pollination and increasing interest in both fields to benefit from the advances of the other. In this article, we review fundamental concepts of causal inference and relate them to crucial open problems of machine learning, including transfer and generalization, thereby assaying how causality can contribute to modern machine learning research. This also applies in the opposite direction: we note that most work in causality starts from the premise that the causal variables are given. A central problem for AI and causality is, thus, causal representation learning, that is, the discovery of high-level causal variables from low-level observations. Finally, we delineate some implications of causality for machine learning and propose key research areas at the intersection of both communities.},
  author       = {Scholkopf, Bernhard and Locatello, Francesco and Bauer, Stefan and Ke, Nan Rosemary and Kalchbrenner, Nal and Goyal, Anirudh and Bengio, Yoshua},
  issn         = {1558-2256},
  journal      = {Proceedings of the IEEE},
  keywords     = {Electrical and Electronic Engineering},
  number       = {5},
  pages        = {612--634},
  publisher    = {Institute of Electrical and Electronics Engineers},
  title        = {{Toward causal representation learning}},
  doi          = {10.1109/jproc.2021.3058954},
  volume       = {109},
  year         = {2021},
}

@inproceedings{14176,
  abstract     = {Intensive care units (ICU) are increasingly looking towards machine learning for methods to provide online monitoring of critically ill patients. In machine learning, online monitoring is often formulated as a supervised learning problem. Recently, contrastive learning approaches have demonstrated promising improvements over competitive supervised benchmarks. These methods rely on well-understood data augmentation techniques developed for image data which do not apply to online monitoring. In this work, we overcome this limitation by
supplementing time-series data augmentation techniques with a novel contrastive
learning objective which we call neighborhood contrastive learning (NCL). Our objective explicitly groups together contiguous time segments from each patient while maintaining state-specific information. Our experiments demonstrate a marked improvement over existing work applying contrastive methods to medical time-series.},
  author       = {Yèche, Hugo and Dresdner, Gideon and Locatello, Francesco and Hüser, Matthias and Rätsch, Gunnar},
  booktitle    = {Proceedings of 38th International Conference on Machine Learning},
  location     = {Virtual},
  pages        = {11964--11974},
  publisher    = {ML Research Press},
  title        = {{Neighborhood contrastive learning applied to online patient monitoring}},
  volume       = {139},
  year         = {2021},
}

@inproceedings{14177,
  abstract     = {The focus of disentanglement approaches has been on identifying independent factors of variation in data. However, the causal variables underlying real-world observations are often not statistically independent. In this work, we bridge the gap to real-world scenarios by analyzing the behavior of the most prominent disentanglement approaches on correlated data in a large-scale empirical study (including 4260 models). We show and quantify that systematically induced correlations in the dataset are being learned and reflected in the latent representations, which has implications for downstream applications of disentanglement such as fairness. We also demonstrate how to resolve these latent correlations, either using weak supervision during
training or by post-hoc correcting a pre-trained model with a small number of labels.},
  author       = {Träuble, Frederik and Creager, Elliot and Kilbertus, Niki and Locatello, Francesco and Dittadi, Andrea and Goyal, Anirudh and Schölkopf, Bernhard and Bauer, Stefan},
  booktitle    = {Proceedings of the 38th International Conference on Machine Learning},
  location     = {Virtual},
  pages        = {10401--10412},
  publisher    = {ML Research Press},
  title        = {{On disentangled representations learned from correlated data}},
  volume       = {139},
  year         = {2021},
}

@inproceedings{14178,
  abstract     = {Learning meaningful representations that disentangle the underlying structure of the data generating process is considered to be of key importance in machine learning. While disentangled representations were found to be useful for diverse tasks such as abstract reasoning and fair classification, their scalability and real-world impact remain questionable. We introduce a new high-resolution dataset with 1M simulated images and over 1,800 annotated real-world images of the same setup. In contrast to previous work, this new dataset exhibits correlations, a complex underlying structure, and allows to evaluate transfer to unseen simulated and real-world settings where the encoder i) remains in distribution or ii) is out of distribution. We propose new architectures in order to scale disentangled representation learning to realistic high-resolution settings and conduct a large-scale empirical study of disentangled representations on this dataset. We observe that disentanglement is a good predictor for out-of-distribution (OOD) task performance.},
  author       = {Dittadi, Andrea and Träuble, Frederik and Locatello, Francesco and Wüthrich, Manuel and Agrawal, Vaibhav and Winther, Ole and Bauer, Stefan and Schölkopf, Bernhard},
  booktitle    = {The Ninth International Conference on Learning Representations},
  location     = {Virtual},
  title        = {{On the transfer of disentangled representations in realistic settings}},
  year         = {2021},
}

@inproceedings{14179,
  abstract     = {Self-supervised representation learning has shown remarkable success in a number of domains. A common practice is to perform data augmentation via hand-crafted transformations intended to leave the semantics of the data invariant. We seek to understand the empirical success of this approach from a theoretical perspective. We formulate the augmentation process as a latent variable model by postulating a partition of the latent representation into a content component, which is assumed invariant to augmentation, and a style component, which is allowed to change. Unlike prior work on disentanglement and independent component analysis, we allow for both nontrivial statistical and causal dependencies in the latent space. We study the identifiability of the latent representation based on pairs of views of the observations and prove sufficient conditions that allow us to identify the invariant content partition up to an invertible mapping in both generative and discriminative settings. We find numerical simulations with dependent latent variables are consistent with our theory. Lastly, we introduce Causal3DIdent, a dataset of high-dimensional, visually complex images with rich causal dependencies, which we use to study the effect of data augmentations performed in practice.},
  author       = {Kügelgen, Julius von and Sharma, Yash and Gresele, Luigi and Brendel, Wieland and Schölkopf, Bernhard and Besserve, Michel and Locatello, Francesco},
  booktitle    = {Advances in Neural Information Processing Systems},
  isbn         = {9781713845393},
  location     = {Virtual},
  pages        = {16451--16467},
  title        = {{Self-supervised learning with data augmentations provably isolates content from style}},
  volume       = {34},
  year         = {2021},
}

@inproceedings{14180,
  abstract     = {Modern neural network architectures can leverage large amounts of data to generalize well within the training distribution. However, they are less capable of systematic generalization to data drawn from unseen but related distributions, a feat that is hypothesized to require compositional reasoning and reuse of knowledge. In this work, we present Neural Interpreters, an architecture that factorizes inference in a self-attention network as a system of modules, which we call \emph{functions}. Inputs to the model are routed through a sequence of functions in a way that is end-to-end learned. The proposed architecture can flexibly compose computation along width and depth, and lends itself well to capacity extension after training. To demonstrate the versatility of Neural Interpreters, we evaluate it in two distinct settings: image classification and visual abstract reasoning on Raven Progressive Matrices. In the former, we show that Neural Interpreters perform on par with the vision transformer using fewer parameters, while being transferrable to a new task in a sample efficient manner. In the latter, we find that Neural Interpreters are competitive with respect to the state-of-the-art in terms of systematic generalization. },
  author       = {Rahaman, Nasim and Gondal, Muhammad Waleed and Joshi, Shruti and Gehler, Peter and Bengio, Yoshua and Locatello, Francesco and Schölkopf, Bernhard},
  booktitle    = {Advances in Neural Information Processing Systems},
  isbn         = {9781713845393},
  location     = {Virtual},
  pages        = {10985--10998},
  title        = {{Dynamic inference with neural interpreters}},
  volume       = {34},
  year         = {2021},
}

@inproceedings{14181,
  abstract     = {Variational Inference makes a trade-off between the capacity of the variational family and the tractability of finding an approximate posterior distribution. Instead, Boosting Variational Inference allows practitioners to obtain increasingly good posterior approximations by spending more compute. The main obstacle to widespread adoption of Boosting Variational Inference is the amount of resources necessary to improve over a strong Variational Inference baseline. In our work, we trace this limitation back to the global curvature of the KL-divergence. We characterize how the global curvature impacts time and memory consumption, address the problem with the notion of local curvature, and provide a novel approximate backtracking algorithm for estimating local curvature. We give new theoretical convergence rates for our algorithms and provide experimental validation on synthetic and real-world datasets.},
  author       = {Dresdner, Gideon and Shekhar, Saurav and Pedregosa, Fabian and Locatello, Francesco and Rätsch, Gunnar},
  booktitle    = {Proceedings of the Thirtieth International Joint Conference on Artificial Intelligence},
  location     = {Montreal, Canada},
  pages        = {2337--2343},
  publisher    = {International Joint Conferences on Artificial Intelligence},
  title        = {{Boosting variational inference with locally adaptive step-sizes}},
  doi          = {10.24963/ijcai.2021/322},
  year         = {2021},
}

@inproceedings{14182,
  abstract     = {When machine learning systems meet real world applications, accuracy is only
one of several requirements. In this paper, we assay a complementary
perspective originating from the increasing availability of pre-trained and
regularly improving state-of-the-art models. While new improved models develop
at a fast pace, downstream tasks vary more slowly or stay constant. Assume that
we have a large unlabelled data set for which we want to maintain accurate
predictions. Whenever a new and presumably better ML models becomes available,
we encounter two problems: (i) given a limited budget, which data points should
be re-evaluated using the new model?; and (ii) if the new predictions differ
from the current ones, should we update? Problem (i) is about compute cost,
which matters for very large data sets and models. Problem (ii) is about
maintaining consistency of the predictions, which can be highly relevant for
downstream applications; our demand is to avoid negative flips, i.e., changing
correct to incorrect predictions. In this paper, we formalize the Prediction
Update Problem and present an efficient probabilistic approach as answer to the
above questions. In extensive experiments on standard classification benchmark
data sets, we show that our method outperforms alternative strategies along key
metrics for backward-compatible prediction updates.},
  author       = {Träuble, Frederik and Kügelgen, Julius von and Kleindessner, Matthäus and Locatello, Francesco and Schölkopf, Bernhard and Gehler, Peter},
  booktitle    = {35th Conference on Neural Information Processing Systems},
  isbn         = {9781713845393},
  location     = {Virtual},
  pages        = {116--128},
  title        = {{Backward-compatible prediction updates: A probabilistic approach}},
  volume       = {34},
  year         = {2021},
}

@unpublished{14221,
  abstract     = {The world is structured in countless ways. It may be prudent to enforce corresponding structural properties to a learning algorithm's solution, such as incorporating prior beliefs, natural constraints, or causal structures. Doing so may translate to faster, more accurate, and more flexible models, which may directly relate to real-world impact. In this dissertation, we consider two different research areas that concern structuring a learning algorithm's solution: when the structure is known and when it has to be discovered.},
  author       = {Locatello, Francesco},
  booktitle    = {arXiv},
  title        = {{Enforcing and discovering structure in machine learning}},
  doi          = {10.48550/arXiv.2111.13693},
  year         = {2021},
}

@unpublished{14278,
  abstract     = {The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral rigidity of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish rigidity for almost all of them.},
  author       = {Koval, Illya},
  booktitle    = {arXiv},
  title        = {{Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse}},
  doi          = {10.48550/ARXIV.2111.12171},
  year         = {2021},
}

@article{9002,
  abstract     = { We prove that, for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of encoding and decoding. Our proof is based on the construction and analysis of binary polar codes with large kernels. When communicating reliably at rates within ε>0 of capacity, the code length n often scales as O(1/εμ), where the constant μ is called the scaling exponent. It is known that the optimal scaling exponent is μ=2, and it is achieved by random linear codes. The scaling exponent of conventional polar codes (based on the 2×2 kernel) on the BEC is μ=3.63. This falls far short of the optimal scaling guaranteed by random codes. Our main contribution is a rigorous proof of the following result: for the BEC, there exist ℓ×ℓ binary kernels, such that polar codes constructed from these kernels achieve scaling exponent μ(ℓ) that tends to the optimal value of 2 as ℓ grows. We furthermore characterize precisely how large ℓ needs to be as a function of the gap between μ(ℓ) and 2. The resulting binary codes maintain the recursive structure of conventional polar codes, and thereby achieve construction complexity O(n) and encoding/decoding complexity O(nlogn).},
  author       = {Fazeli, Arman and Hassani, Hamed and Mondelli, Marco and Vardy, Alexander},
  issn         = {1557-9654},
  journal      = {IEEE Transactions on Information Theory},
  number       = {9},
  pages        = {5693--5710},
  publisher    = {IEEE},
  title        = {{Binary linear codes with optimal scaling: Polar codes with large kernels}},
  doi          = {10.1109/TIT.2020.3038806},
  volume       = {67},
  year         = {2021},
}

@article{9005,
  abstract     = {Studies on the experimental realization of two-dimensional anyons in terms of quasiparticles have been restricted, so far, to only anyons on the plane. It is known, however, that the geometry and topology of space can have significant effects on quantum statistics for particles moving on it. Here, we have undertaken the first step toward realizing the emerging fractional statistics for particles restricted to move on the sphere instead of on the plane. We show that such a model arises naturally in the context of quantum impurity problems. In particular, we demonstrate a setup in which the lowest-energy spectrum of two linear bosonic or fermionic molecules immersed in a quantum many-particle environment can coincide with the anyonic spectrum on the sphere. This paves the way toward the experimental realization of anyons on the sphere using molecular impurities. Furthermore, since a change in the alignment of the molecules corresponds to the exchange of the particles on the sphere, such a realization reveals a novel type of exclusion principle for molecular impurities, which could also be of use as a powerful technique to measure the statistics parameter. Finally, our approach opens up a simple numerical route to investigate the spectra of many anyons on the sphere. Accordingly, we present the spectrum of two anyons on the sphere in the presence of a Dirac monopole field.},
  author       = {Brooks, Morris and Lemeshko, Mikhail and Lundholm, D. and Yakaboylu, Enderalp},
  issn         = {10797114},
  journal      = {Physical Review Letters},
  number       = {1},
  publisher    = {American Physical Society},
  title        = {{Molecular impurities as a realization of anyons on the two-sphere}},
  doi          = {10.1103/PhysRevLett.126.015301},
  volume       = {126},
  year         = {2021},
}

@article{9006,
  abstract     = {Cytoplasm is a gel-like crowded environment composed of various macromolecules, organelles, cytoskeletal networks, and cytosol. The structure of the cytoplasm is highly organized and heterogeneous due to the crowding of its constituents and their effective compartmentalization. In such an environment, the diffusive dynamics of the molecules are restricted, an effect that is further amplified by clustering and anchoring of molecules. Despite the crowded nature of the cytoplasm at the microscopic scale, large-scale reorganization of the cytoplasm is essential for important cellular functions, such as cell division and polarization. How such mesoscale reorganization of the cytoplasm is achieved, especially for large cells such as oocytes or syncytial tissues that can span hundreds of micrometers in size, is only beginning to be understood. In this review, we will discuss recent advances in elucidating the molecular, cellular, and biophysical mechanisms by which the cytoskeleton drives cytoplasmic reorganization across different scales, structures, and species.},
  author       = {Shamipour, Shayan and Caballero Mancebo, Silvia and Heisenberg, Carl-Philipp J},
  issn         = {18781551},
  journal      = {Developmental Cell},
  number       = {2},
  pages        = {P213--226},
  publisher    = {Elsevier},
  title        = {{Cytoplasm's got moves}},
  doi          = {10.1016/j.devcel.2020.12.002},
  volume       = {56},
  year         = {2021},
}

@article{9009,
  abstract     = {Recent advancements in live cell imaging technologies have identified the phenomenon of intracellular propagation of late apoptotic events, such as cytochrome c release and caspase activation. The mechanism, prevalence, and speed of apoptosis propagation remain unclear. Additionally, no studies have demonstrated propagation of the pro-apoptotic protein, BAX. To evaluate the role of BAX in intracellular apoptotic propagation, we used high speed live-cell imaging to visualize fluorescently tagged-BAX recruitment to mitochondria in four immortalized cell lines. We show that propagation of mitochondrial BAX recruitment occurs in parallel to cytochrome c and SMAC/Diablo release and is affected by cellular morphology, such that cells with processes are more likely to exhibit propagation. The initiation of propagation events is most prevalent in the distal tips of processes, while the rate of propagation is influenced by the 2-dimensional width of the process. Propagation was rarely observed in the cell soma, which exhibited near synchronous recruitment of BAX. Propagation velocity is not affected by mitochondrial volume in segments of processes, but is negatively affected by mitochondrial density. There was no evidence of a propagating wave of increased levels of intracellular calcium ions. Alternatively, we did observe a uniform increase in superoxide build-up in cellular mitochondria, which was released as a propagating wave simultaneously with the propagating recruitment of BAX to the mitochondrial outer membrane.},
  author       = {Grosser, Joshua A. and Maes, Margaret E and Nickells, Robert W.},
  issn         = {1573-675X},
  journal      = {Apoptosis},
  number       = {2},
  pages        = {132--145},
  publisher    = {Springer Nature},
  title        = {{Characteristics of intracellular propagation of mitochondrial BAX recruitment during apoptosis}},
  doi          = {10.1007/s10495-020-01654-w},
  volume       = {26},
  year         = {2021},
}

@article{9010,
  abstract     = {Availability of the essential macronutrient nitrogen in soil plays a critical role in plant growth, development, and impacts agricultural productivity. Plants have evolved different strategies for sensing and responding to heterogeneous nitrogen distribution. Modulation of root system architecture, including primary root growth and branching, is among the most essential plant adaptions to ensure adequate nitrogen acquisition. However, the immediate molecular pathways coordinating the adjustment of root growth in response to distinct nitrogen sources, such as nitrate or ammonium, are poorly understood. Here, we show that growth as manifested by cell division and elongation is synchronized by coordinated auxin flux between two adjacent outer tissue layers of the root. This coordination is achieved by nitrate‐dependent dephosphorylation of the PIN2 auxin efflux carrier at a previously uncharacterized phosphorylation site, leading to subsequent PIN2 lateralization and thereby regulating auxin flow between adjacent tissues. A dynamic computer model based on our experimental data successfully recapitulates experimental observations. Our study provides mechanistic insights broadening our understanding of root growth mechanisms in dynamic environments.},
  author       = {Ötvös, Krisztina and Marconi, Marco and Vega, Andrea and O’Brien, Jose and Johnson, Alexander J and Abualia, Rashed and Antonielli, Livio and Montesinos López, Juan C and Zhang, Yuzhou and Tan, Shutang and Cuesta, Candela and Artner, Christina and Bouguyon, Eleonore and Gojon, Alain and Friml, Jiří and Gutiérrez, Rodrigo A. and Wabnik, Krzysztof T and Benková, Eva},
  issn         = {14602075},
  journal      = {EMBO Journal},
  number       = {3},
  publisher    = {Embo Press},
  title        = {{Modulation of plant root growth by nitrogen source-defined regulation of polar auxin transport}},
  doi          = {10.15252/embj.2020106862},
  volume       = {40},
  year         = {2021},
}

@article{9020,
  abstract     = {We study dynamics and thermodynamics of ion transport in narrow, water-filled channels, considered as effective 1D Coulomb systems. The long range nature of the inter-ion interactions comes about due to the dielectric constants mismatch between the water and the surrounding medium, confining the electric filed to stay mostly within the water-filled channel. Statistical mechanics of such Coulomb systems is dominated by entropic effects which may be accurately accounted for by mapping onto an effective quantum mechanics. In presence of multivalent ions the corresponding quantum mechanics appears to be non-Hermitian. In this review we discuss a framework for semiclassical calculations for the effective non-Hermitian Hamiltonians. Non-Hermiticity elevates WKB action integrals from the real line to closed cycles on a complex Riemann surfaces where direct calculations are not attainable. We circumvent this issue by applying tools from algebraic topology, such as the Picard-Fuchs equation. We discuss how its solutions relate to the thermodynamics and correlation functions of multivalent solutions within narrow, water-filled channels. },
  author       = {Gulden, Tobias and Kamenev, Alex},
  issn         = {1099-4300},
  journal      = {Entropy},
  number       = {1},
  publisher    = {MDPI},
  title        = {{Dynamics of ion channels via non-hermitian quantum mechanics}},
  doi          = {10.3390/e23010125},
  volume       = {23},
  year         = {2021},
}

@phdthesis{9022,
  abstract     = {In the first part of the thesis we consider Hermitian random matrices. Firstly, we consider sample covariance matrices XX∗ with X having independent identically distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences of linear statistics of XX∗ and its minor after removing the first column of X. Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics near cusp singularities of the limiting density of states are universal and that they form a Pearcey process. Since the limiting eigenvalue distribution admits only square root (edge) and cubic root (cusp) singularities, this concludes the third and last remaining case of the Wigner-Dyson-Mehta universality conjecture. The main technical ingredients are an optimal local law at the cusp, and the proof of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp regime.
In the second part we consider non-Hermitian matrices X with centred i.i.d. entries. We normalise the entries of X to have variance N −1. It is well known that the empirical eigenvalue density converges to the uniform distribution on the unit disk (circular law). In the first project, we prove universality of the local eigenvalue statistics close to the edge of the spectrum. This is the non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck flow for very long time
(up to t = +∞). In the second project, we consider linear statistics of eigenvalues for macroscopic test functions f in the Sobolev space H2+ϵ and prove their convergence to the projection of the Gaussian Free Field on the unit disk. We prove this result for non-Hermitian matrices with real or complex entries. The main technical ingredients are: (i) local law for products of two resolvents at different spectral parameters, (ii) analysis of correlated Dyson Brownian motions.
In the third and final part we discuss the mathematically rigorous application of supersymmetric techniques (SUSY ) to give a lower tail estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we use superbosonisation formula to give an integral representation of the resolvent of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex and real case, respectively. The rigorous analysis of these integrals is quite challenging since simple saddle point analysis cannot be applied (the main contribution comes from a non-trivial manifold). Our result
improves classical smoothing inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality for i.i.d. non-Hermitian matrices.},
  author       = {Cipolloni, Giorgio},
  issn         = {2663-337X},
  pages        = {380},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Fluctuations in the spectrum of random matrices}},
  doi          = {10.15479/AT:ISTA:9022},
  year         = {2021},
}

@article{9036,
  abstract     = {In this short note, we prove that the square root of the quantum Jensen-Shannon divergence is a true metric on the cone of positive matrices, and hence in particular on the quantum state space.},
  author       = {Virosztek, Daniel},
  issn         = {0001-8708},
  journal      = {Advances in Mathematics},
  keywords     = {General Mathematics},
  number       = {3},
  publisher    = {Elsevier},
  title        = {{The metric property of the quantum Jensen-Shannon divergence}},
  doi          = {10.1016/j.aim.2021.107595},
  volume       = {380},
  year         = {2021},
}