Patrick Lutz

Peter Cholak, Marianna Csörnyei, Neil Lutz et al.

We introduce a new technique for proving bounds on the Kolmogorov complexity of geometric objects in Euclidean space, such as points and lines. We apply this technique to prove two theorems on algorithmic information theory, both of which have consequences for well-known problems in geometric measure theory. First, we show that for any point $x$ in the plane and any other point $y$ sufficiently independent of $x$, the distance between $x$ and $y$ retains at least half the complexity of the original point $x$. By the point-to-set principle of J. Lutz and N. Lutz, this yields an improved lower bound on the Hausdorff dimension of pinned distance sets, a topic closely related to Falconer's distance set conjecture. Second, we prove an analogous result for orthogonal projections: for any point $x$ in the plane and any line through the origin which is sufficiently independent of $x$, the projection of $x$ onto that line retains at least half the complexity of $x$. As a consequence, we obtain a generalization of a theorem of Bourgain on exceptional sets for orthogonal projections.

Patrick Lutz, Ludovic Arnould, Claire Boyer et al.

Dedicated neural network (NN) architectures have been designed to handle specific data types (such as CNN for images or RNN for text), which ranks them among state-of-the-art methods for dealing with these data. Unfortunately, no architecture has been found for dealing with tabular data yet, for which tree ensemble methods (tree boosting, random forests) usually show the best predictive performances. In this work, we propose a new sparse initialization technique for (potentially deep) multilayer perceptrons (MLP): we first train a tree-based procedure to detect feature interactions and use the resulting information to initialize the network, which is subsequently trained via standard stochastic gradient strategies. Numerical experiments on several tabular data sets show that this new, simple and easy-to-use method is a solid concurrent, both in terms of generalization capacity and computation time, to default MLP initialization and even to existing complex deep learning solutions. In fact, this wise MLP initialization raises the resulting NN methods to the level of a valid competitor to gradient boosting when dealing with tabular data. Besides, such initializations are able to preserve the sparsity of weights introduced in the first layers of the network through training. This fact suggests that this new initializer operates an implicit regularization during the NN training, and emphasizes that the first layers act as a sparse feature extractor (as for convolutional layers in CNN).

Patrick Lutz, Aditya Gangrade, Hadi Daneshmand et al.

We train a linear attention transformer on millions of masked-block matrix completion tasks: each prompt is masked low-rank matrix whose missing block may be (i) a scalar prediction target or (ii) an unseen kernel slice of Nyström extrapolation. The model sees only input-output pairs and a mean-squared loss; it is given no normal equations, no handcrafted iterations, and no hint that the tasks are related. Surprisingly, after training, algebraic unrolling reveals the same parameter-free update rule across three distinct computational regimes (full visibility, rank-limited updates, and distributed computation). We prove that this rule achieves second-order convergence on full-batch problems, cuts distributed iteration complexity, and remains accurate with rank-limited attention. Thus, a transformer trained solely to patch missing blocks implicitly discovers a unified, resource-adaptive iterative solver spanning prediction, estimation, and Nyström extrapolation, highlighting a powerful capability of in-context learning.