One of the latest tools NVIDIA has announced at IFA 2026 is called Personal AI Router or PAIR, a free and open-source tool that can distribute AI workloads across local PCs. "More than half of US households have two or more PCs," NVIDIA writes in its announcement, and apparently, they mostly sit idle throughout the day. While AI agents already break complex tasks into smaller jobs that can be done at the same time, they could be competing for the same GPU anyway. In other words, more complex tasks would still take some time to finish. PAIR prevents performance slowdowns by finding PCs on a local network and then routing task requests to the system, which is currently running idle and has the capacity to accomplish them more quickly.

What is Iran's Pickaxe Mountain? Six months into the war with Iran, it is worth taking stock of what has actually changed around the Gulf. After Iran's June 2025 missile strike on Al Udeid Air Base in Qatar, I argued that the years ahead would be more perilous for the Gulf and that deeper defence cooperation and integration would eventually become unavoidable. Earlier this month, I wrote here that the Gulf could no longer rely entirely on the old security toolkit and would need to build more of a security ecosystem of its own. So far, events have borne out the broad direction of both arguments.

Graphical models have been widely used as parsimonious encoders of assumptions of the underlying causal system and provide a basis for causal inferences. Models encoding stronger constraints tend to require higher expressive power, which are also harder, and sometimes impossible to empirically falsify. In this paper, we introduce two new collections of distributions that include counterfactual quantities which are experimentally accessible under counterfactual randomizations. Correspondingly, we define two new classes of graphical models for encoding empirically testable constraints in these distributions. We further present a sound and complete calculus, based on counterfactual calculus, which licenses inferences in these two new models with rules that are within the empirically falsifiable boundary. Finally, we formulate a hierarchy over several graphical models based on the constraints they encode and study the fundamental trade-off between the expressive power and empirical falsifiability of different models across the hierarchy.

We propose a novel energy function for Dense Associative Memory (DenseAM) networks, the log-sum-ReLU (LSR), inspired by optimal kernel density estimation. Unlike the common log-sum-exponential (LSE) function, LSR is based on the Epanechnikov kernel and enables exact memory retrieval with exponential capacity without requiring exponential separation functions. Moreover, it introduces abundant additional emergent local minima while preserving perfect pattern recovery -- a characteristic previously unseen in DenseAM literature. Empirical results show that LSR energy has significantly more local minima (memories) that have comparable log-likelihood to LSE-based models. Analysis of LSR's emergent memories on image datasets reveals a degree of creativity and novelty, hinting at this method's potential for both large-scale memory storage and generative tasks.

We consider the problem of testing properties of graphs underlying high-dimensional graphical models. We adopt the model of covariance queries introduced by Lugosi, Truszkowski, Velona, and Zwiernik (2021). We study the case when the underlying graph is a tree. The main results of the paper show that, while reconstructing the entire tree may be costly, certain global structural properties can be tested efficiently. In particular, we design randomized tests for global structural properties that use a sub-quadratic number of queries. We develop testing procedures for several fundamental properties, including the number of leaves, the maximum degree, the typical distance, and the diameter of the tree. For each property, we obtain explicit query complexity bounds that depend on the target threshold and tolerance parameters.

How many key-value associations can a $d\times d$ linear memory store? We show that the answer depends not only on the $d^2$ degrees of freedom in the memory matrix, but also on the retrieval criterion. In an isotropic Gaussian model for the stored pairs, we show that top-1 retrieval, where every signal must beat its largest distractor, requires the logarithmic model-size scale $d^2\asymp n\log n$. We prove that the correlation matrix memory construction, which stores associations by superposing key-target outer products, achieves this scale through a sharp phase transition, and that the same scaling is necessary for any linear memory. Thus the logarithm is the intrinsic extreme-value price of winner-take-all decoding. We next consider listwise retrieval, where the correct target need not be the unique top-scoring item but should remain among the strongest candidates. To formalize this regime, we propose the Tail-Average Margin (TAM), a convex upper-tail criterion that certifies inclusion of the correct target in a controlled candidate list. Under this listwise retrieval criterion, the capacity follows the quadratic scale $d^2\asymp n$. At load $n/d^2\toα$, we develop an exact asymptotic theory for the TAM empirical-risk minimizer through a two-parameter scalar variational principle. The theory has a rich phenomenology: in the ridgeless limit it yields a closed-form critical load separating satisfiable and unsatisfiable phases, and it predicts the limiting laws of true scores, competitor scores, margins, and percentile profiles. Finally, a small-tail extrapolation further leads to the conjectural sharp top-1 threshold $d^2\sim 2n\log n$.

We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.

Graph data are inherently complex and heterogeneous, leading to a high natural diversity of distributional shifts. However, it remains unclear how to build machine learning architectures that generalize to the complex distributional shifts naturally occurring in the real world. Here, we develop GraphMETRO, a Graph Neural Network architecture that models natural diversity and captures complex distributional shifts. GraphMETRO employs a Mixture-of-Experts (MoE) architecture with a gating model and multiple expert models, where each expert model targets a specific distributional shift to produce a referential representation w.r.t. a reference model, and the gating model identifies shift components. Additionally, we design a novel objective that aligns the representations from different expert models to ensure reliable optimization. GraphMETRO achieves state-of-the-art results on four datasets from the GOOD benchmark, which is comprised of complex and natural real-world distribution shifts, improving by 67% and 4.2% on the WebKB and Twitch datasets.