For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily We are now in a position to present the main theorem of this subsection. To avoid repetition, we defer further discussions on the rates observed in Theorem A.1 to Remark 2.7 where a holistic In fact, by Proposition 1.1, there exists an optimal transport map Based on (B.2), the natural plug-in estimator of ρ Suppose that the same assumptions from Theorem 2.2 hold. B.2 Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion Proposition B.2 shows that the test based on Further, when the sampling distribution is fixed, Proposition B.2 shows that In the following result (see Appendix C.2 for a proof), we show that if This section is devoted to proving our main results and is organized as follows: In Appendix C.1, we Further by Lemma D.2, we also have: ϕ Note that (C.10) immediately yields the following conclusions: S By (1.5) and some simple algebra, the following holds: null null null S Combining the above display with (C.9), we further have: null null null null 1 2 W Combining the above observation with Theorem 2.1, we have: lim sup For the next part, to simplify notation, let us begin with some notation. By using the exponential Markov's inequality coupled with the standard union Now by using [7, Theorem 2.10], we have P (B We are now in a position to complete the proof of Theorem 2.2 using steps I-III. Therefore, it is now enough to bound the right hand side of (C.17).

For the moment, it is worth noting that such sets of functions (e.g., Haar wavelets, Daubechies wavelets) are readily We are now in a position to present the main theorem of this subsection. To avoid repetition, we defer further discussions on the rates observed in Theorem A.1 to Remark 2.7 where a holistic In fact, by Proposition 1.1, there exists an optimal transport map Based on (B.2), the natural plug-in estimator of ρ Suppose that the same assumptions from Theorem 2.2 hold. B.2 Nonparametric independence testing: Optimal transport based Hilbert-Schmidt independence criterion Proposition B.2 shows that the test based on Further, when the sampling distribution is fixed, Proposition B.2 shows that In the following result (see Appendix C.2 for a proof), we show that if This section is devoted to proving our main results and is organized as follows: In Appendix C.1, we Further by Lemma D.2, we also have: ϕ Note that (C.10) immediately yields the following conclusions: S By (1.5) and some simple algebra, the following holds: null null null S Combining the above display with (C.9), we further have: null null null null 1 2 W Combining the above observation with Theorem 2.1, we have: lim sup For the next part, to simplify notation, let us begin with some notation. By using the exponential Markov's inequality coupled with the standard union Now by using [7, Theorem 2.10], we have P (B We are now in a position to complete the proof of Theorem 2.2 using steps I-III. Therefore, it is now enough to bound the right hand side of (C.17).

We now show why this tell us to pick the all-ones vector for SM Kernels: Corollary 4. So, by Lemma 1, we complete the proof. With this reduction in place, we move onto consider the means and lengthscales of our kernel. C for all ξ, proven below. C.1 Proof for the Matrix Case First, we introduce the matrix version of the ridge leverage function, first introduced in [AM15]: Definition 3. F or a matrix A R A + εI) Then we move onto the theorem we want to prove: 16 Theorem 5. We bound these two terms separately, starting with the latter. Hence, by Markov's inequality, we have null( S (A C.2 Proof for the Operator Case We start with preliminary definitions for randomized operator analysis.

Integrating brain imaging data with clinical reports offers a valuable opportunity to leverage complementary multimodal information for more effective and timely diagnosis in practical clinical settings. This approach has gained significant attention in brain disorder research, yet a key challenge remains: how to effectively link objective imaging data with subjective text-based reports, such as doctors' notes. In this work, we propose a novel framework that aligns brain connectomes with clinical reports in a shared cross-modal latent space at both the subject and connectome levels, thereby enhancing representation learning. The key innovation of our approach is that we treat brain subnetworks as tokens of imaging data, rather than raw image patches, to align with word tokens in clinical reports. This enables a more efficient identification of system-level associations between neuroimaging findings and clinical observations, which is critical since brain disorders often manifest as network-level abnormalities rather than isolated regional alterations. We applied our method to mild cognitive impairment (MCI) using the ADNI dataset. Our approach not only achieves state-of-the-art predictive performance but also identifies clinically meaningful connectome-text pairs, offering new insights into the early mechanisms of Alzheimer's disease and supporting the development of clinically useful multimodal biomarkers.

Compliant mechanisms have significant potential in precision applications due to their ability to guide motion without contact. However, an inherent vulnerability to fatigue and mechanical failure has hindered the translation of compliant mechanisms to real-world applications. This is particularly challenging in service environments where loading is complex and uncertain, and the cost of failure is high. In such cases, mechanical hard stops are critical to prevent yielding and buckling. Conventional hard-stop designs, which rely on stacking single-DOF limits, must be overly restrictive in multi-DOF space to guarantee safety in the presence of unknown loads. In this study, we present a systematic design synthesis method to guarantee overload protection in compliant mechanisms by integrating coupled multi-DOF motion limits within a single pair of compact hard-stop surfaces. Specifically, we introduce a theoretical and practical framework for optimizing the contact surface geometry to maximize the mechanism's multi-DOF working space while still ensuring that the mechanism remains within its elastic regime. We apply this synthesis method to a case study of a caged-hinge mechanism for orthopaedic implants, and provide numerical and experimental validation that the derived design offers reliable protection against fatigue, yielding, and buckling. This work establishes a foundation for precision hard-stop design in compliant systems operating under uncertain loads, which is a crucial step toward enabling the application of compliant mechanisms in real-world systems.

In a variety of physically relevant settings for learning from quantum data, designing protocols that can computationally efficiently extract information remains largely an art, and there are important cases where we believe this to be impossible, that is, where there is an information-computation gap. While there is a large array of tools in the classical literature for giving evidence for average-case hardness of statistical inference problems, the corresponding tools in the quantum literature are far more limited. One such framework in the classical literature, the low-degree method, makes predictions about hardness of inference problems based on the failure of estimators given by low-degree polynomials. In this work, we extend this framework to the quantum setting. We establish a general connection between state designs and low-degree hardness. We use this to obtain the first information-computation gaps for learning Gibbs states of random, sparse, non-local Hamiltonians. We also use it to prove hardness for learning random shallow quantum circuit states in a challenging model where states can be measured in adaptively chosen bases. To our knowledge, the ability to model adaptivity within the low-degree framework was open even in classical settings. In addition, we also obtain a low-degree hardness result for quantum error mitigation against strategies with single-qubit measurements. We define a new quantum generalization of the planted biclique problem and identify the threshold at which this problem becomes computationally hard for protocols that perform local measurements. Interestingly, the complexity landscape for this problem shifts when going from local measurements to more entangled single-copy measurements. We show average-case hardness for the "standard" variant of Learning Stabilizers with Noise and for agnostically learning product states.

Successful collaboration requires team members to stay aligned, especially in complex sequential tasks. Team members must dynamically coordinate which subtasks to perform and in what order. However, real-world constraints like partial observability and limited communication bandwidth often lead to suboptimal collaboration. Even among expert teams, the same task can be executed in multiple ways. To develop multi-agent systems and human-AI teams for such tasks, we are interested in data-driven learning of multimodal team behaviors. Multi-Agent Imitation Learning (MAIL) provides a promising framework for data-driven learning of team behavior from demonstrations, but existing methods struggle with heterogeneous demonstrations, as they assume that all demonstrations originate from a single team policy. Hence, in this work, we introduce DTIL: a hierarchical MAIL algorithm designed to learn multimodal team behaviors in complex sequential tasks. DTIL represents each team member with a hierarchical policy and learns these policies from heterogeneous team demonstrations in a factored manner. By employing a distribution-matching approach, DTIL mitigates compounding errors and scales effectively to long horizons and continuous state representations. Experimental results show that DTIL outperforms MAIL baselines and accurately models team behavior across a variety of collaborative scenarios.

The classical $\textit{Procrustes}$ problem is to find a rigid motion (orthogonal transformation and translation) that best aligns two given point-sets in the least-squares sense. The $\textit{Robust Procrustes}$ problem is an important variant, in which a power-1 objective is used instead of least squares to improve robustness to outliers. While the optimal solution of the least-squares problem can be easily computed in closed form, dating back to Sch\"onemann (1966), no such solution is known for the power-1 problem. In this paper we propose a novel convex relaxation for the Robust Procrustes problem. Our relaxation enjoys several theoretical and practical advantages: Theoretically, we prove that our method provides a $\sqrt{2}$-factor approximation to the Robust Procrustes problem, and that, under appropriate assumptions, it exactly recovers the true rigid motion from point correspondences contaminated by outliers. In practice, we find in numerical experiments on both synthetic and real robust Procrustes problems, that our method performs similarly to the standard Iteratively Reweighted Least Squares (IRLS). However the convexity of our algorithm allows incorporating additional convex penalties, which are not readily amenable to IRLS. This turns out to be a substantial advantage, leading to improved results in high-dimensional problems, including non-rigid shape alignment and semi-supervised interlingual word translation.