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V. Matache (J. Operator Theory 73(1):243--264, 2015) raised an open problem about characterizing composition operators $C_{\phi}$ on the Hardy space $H^2$ and nonzero singular measures $\mu_1$, $\mu_2$ on the unit circle such that $C_{\phi}({S_{\mu_1}} H^2)\subseteq {S_{\mu_2}} H^2,$ where $S_{\mu_i}$ denotes the singular inner function corresponding to the measure $\mu_i,i=1,2$. In this article, we consider this problem in a more general setting. We characterize holomorphic self maps $\phi$ of the unit disk $\mathbb{D}$ and inner functions $\theta_1, \theta_2$ such that $C_{\phi}(\theta_1 H^p)\subseteq \theta_2 H^p,$ for $p>0$. Emphasis is given to Blaschke products and singular inner functions as a special case. We also give an another measure-theoretic characterization to above question when $\phi$ is an elliptic automorphism. For a given Blaschke product $\theta$, we discuss about finding all self maps $\phi$ such that $\theta H^p$ is invariant under $C_\phi$., Comment: 15 pages

Purpose: Estimation of multi-compartment intravoxel flow in fD in ml/100g/min with multi-b-value diffusion weighted imaging and a multi-Gaussian model in the kidneys. Theory and Methods: A multi-Gaussian model of intravoxel flow using water transport time to quantify fD is presented and simulated. Multi-compartment anisotropic DWI signal is simulated analyzed with (1) a rigid bi-exponential, (2) a rigid tri-exponential, and (3) diffusion spectrum imaging model of intravoxel incoherent motion (spectral diffusion). The application is demonstrated in a two-center study of 54 kidney allografts with 9 b-value advanced DWI that were split by function (CKD-EPI 2021 eGFR<45ml/min/1.73m2) and fibrosis (Banff 2017 interstitial fibrosis and tubular atrophy score 0-6). Results: Spectral diffusion demonstrated strong correlation to truth for simulated three-compartment anisotropic diffusion (y=1.08x+0.1, R2=0.71) and two-compartment anisotropic diffusion (y=0.91x+0.6, R2=0.74), outperforming rigid models in cases of variable compartment number. Use of a fixed regularization parameter set to {\lambda}=0.1 increased computation up to 208-fold and agreed with voxel-wise cross-validated regularization (concordance correlation coefficient=0.99). Spectral diffusion of renal allografts showed significant increase in tissue parenchyma compartment fD (f-stat=3.86, p=0.02). Tubular fD was significantly decreased in allografts with impaired function (Mann-Whitney Utest t-stat=-2.14, p=0.04). Conclusions: Quantitative multi-compartment intravoxel flow can be estimated in ml/100g/min with fD from multi-Gaussian diffusion, even with moderate anisotropy such as in kidneys. The use of spectral diffusion with a multi-Gaussian model and a fixed regularization parameter shows promise in organs such as the kidney with variable numbers of physiologic compartments.

Modern machine learning models are prone to over-reliance on spurious correlations, which can often lead to poor performance on minority groups. In this paper, we identify surprising and nuanced behavior of finetuned models on worst-group accuracy via comprehensive experiments on four well-established benchmarks across vision and language tasks. We first show that the commonly used class-balancing techniques of mini-batch upsampling and loss upweighting can induce a decrease in worst-group accuracy (WGA) with training epochs, leading to performance no better than without class-balancing. While in some scenarios, removing data to create a class-balanced subset is more effective, we show this depends on group structure and propose a mixture method which can outperform both techniques. Next, we show that scaling pretrained models is generally beneficial for worst-group accuracy, but only in conjuction with appropriate class-balancing. Finally, we identify spectral imbalance in finetuning features as a potential source of group disparities -- minority group covariance matrices incur a larger spectral norm than majority groups once conditioned on the classes. Our results show more nuanced interactions of modern finetuned models with group robustness than was previously known. Our code is available at https://github.com/tmlabonte/revisiting-finetuning.

Robotic weed flaming is a new and environmentally friendly approach to weed removal in the agricultural field. Using a mobile manipulator equipped with a flamethrower, we design a new system and algorithm to enable effective weed flaming, which requires robotic manipulation with a soft and deformable end effector, as the thermal coverage of the flame is affected by dynamic or unknown environmental factors such as gravity, wind, atmospheric pressure, fuel tank pressure, and pose of the nozzle. System development includes overall design, hardware integration, and software pipeline. To enable precise weed removal, the greatest challenge is to detect and predict dynamic flame coverage in real time before motion planning, which is quite different from a conventional rigid gripper in grasping or a spray gun in painting. Based on the images from two onboard infrared cameras and the pose information of the flamethrower nozzle on a mobile manipulator, we propose a new dynamic flame coverage model. The flame model uses a center-arc curve with a Gaussian cross-section model to describe the flame coverage in real time. The experiments have demonstrated the working system and shown that our model and algorithm can achieve a mean average precision (mAP) of more than 76\% in the reprojected images during online prediction., Comment: IROS 2024

Classifying 3D MRI images for early detection of Alzheimer's disease is a critical task in medical imaging. Traditional approaches using Convolutional Neural Networks (CNNs) and Transformers face significant challenges in this domain. CNNs, while effective in capturing local spatial features, struggle with long-range dependencies and often require extensive computational resources for high-resolution 3D data. Transformers, on the other hand, excel in capturing global context but suffer from quadratic complexity in inference time and require substantial memory, making them less efficient for large-scale 3D MRI data. To address these limitations, we propose the use of Vision Mamba, an advanced model based on State Space Models (SSMs), for the classification of 3D MRI images to detect Alzheimer's disease. Vision Mamba leverages dynamic state representations and the selective scan algorithm, allowing it to efficiently capture and retain important spatial information across 3D volumes. By dynamically adjusting state transitions based on input features, Vision Mamba can selectively retain relevant information, leading to more accurate and computationally efficient processing of 3D MRI data. Our approach combines the parallelizable nature of convolutional operations during training with the efficient, recurrent processing of states during inference. This architecture not only improves computational efficiency but also enhances the model's ability to handle long-range dependencies within 3D medical images. Experimental results demonstrate that Vision Mamba outperforms traditional CNN and Transformer models accuracy, making it a promising tool for the early detection of Alzheimer's disease using 3D MRI data., Comment: 12 pages with 5 figures and 3 tables, to be submitted as a book chapter

The classical iteratively reweighted least-squares (IRLS) algorithm aims to recover an unknown signal from linear measurements by performing a sequence of weighted least squares problems, where the weights are recursively updated at each step. Varieties of this algorithm have been shown to achieve favorable empirical performance and theoretical guarantees for sparse recovery and $\ell_p$-norm minimization. Recently, some preliminary connections have also been made between IRLS and certain types of non-convex linear neural network architectures that are observed to exploit low-dimensional structure in high-dimensional linear models. In this work, we provide a unified asymptotic analysis for a family of algorithms that encompasses IRLS, the recently proposed lin-RFM algorithm (which was motivated by feature learning in neural networks), and the alternating minimization algorithm on linear diagonal neural networks. Our analysis operates in a "batched" setting with i.i.d. Gaussian covariates and shows that, with appropriately chosen reweighting policy, the algorithm can achieve favorable performance in only a handful of iterations. We also extend our results to the case of group-sparse recovery and show that leveraging this structure in the reweighting scheme provably improves test error compared to coordinate-wise reweighting., Comment: 25 pages, 3 figures

We study the role of active coupling on the transport properties of homogeneously charged macromolecules in an infinitely dilute solution. An enzyme becomes actively bound to a segment of the macromolecule, exerting an electrostatic force on it. Eventually, thermal fluctuations cause it to become unbound, introducing active coupling into the system. We study the mean-squared displacement (MSD) and find a new scaling regime compared to the thermal counterpart in the presence of hydrodynamic and segment-segment electrostatic interactions. Furthermore, the study of segment-segment equal-time correlation reveals the swelling of the macromolecule. Further, we derive the concentration equation of the macromolecule with active binding and study how the cooperative diffusivity of the macromolecules get modified by its environment, including the macromolecules itself. It turns out that these active fluctuations enhance the effective diffusivity of the macromolecules. The derived closed-form expression for diffusion constant is pertinent to the accurate interpretation of light scattering data in multi-component systems with binding-unbinding equilibria., Comment: 13 pages, 1 figure

The reduction in coral reef densities, characterized by the proliferation of macroalgae, has emerged as a global threat. In this paper, we present a discrete-time coral reef dynamical model that incorporates macroalgae. We explore all ecologically possible equilibrium points for the proposed model. The conditions for the local stability of the interior equilibrium point are analyzed, which represents the coexistence of both coral and macroalgae. Furthermore, we investigate the model's behavior using the center manifold theorem and bifurcation theory. Our analysis reveals that the model undergoes codimension-one bifurcations, specifically period-doubling and Neimark-Sacker bifurcations. To address the chaos resulting from the emergence of the Neimark-Sacker bifurcation, we apply the OGY feedback control method and a hybrid control methodology. Finally, we provide numerical simulations not only to validate the obtained results but also to demonstrate the complex dynamic behaviors that arise. These behaviors include reversal period-doubling bifurcation, period-4, 8, and 24 bubble bifurcations, as well as chaotic behavior.

Overparameterized models that achieve zero training error are observed to generalize well on average, but degrade in performance when faced with data that is under-represented in the training sample. In this work, we study an overparameterized Gaussian mixture model imbued with a spurious feature, and sharply analyze the in-distribution and out-of-distribution test error of a cost-sensitive interpolating solution that incorporates "importance weights". Compared to recent work Wang et al. (2021), Behnia et al. (2022), our analysis is sharp with matching upper and lower bounds, and significantly weakens required assumptions on data dimensionality. Our error characterizations also apply to any choice of importance weights and unveil a novel tradeoff between worst-case robustness to distribution shift and average accuracy as a function of the importance weight magnitude., Comment: A short version of this work will be presented at IEEE ISIT 2024

We study the problem of estimating the stationary mass -- also called the unigram mass -- that is missing from a single trajectory of a discrete-time, ergodic Markov chain. This problem has several applications -- for example, estimating the stationary missing mass is critical for accurately smoothing probability estimates in sequence models. While the classical Good--Turing estimator from the 1950s has appealing properties for i.i.d. data, it is known to be biased in the Markov setting, and other heuristic estimators do not come equipped with guarantees. Operating in the general setting in which the size of the state space may be much larger than the length $n$ of the trajectory, we develop a linear-runtime estimator called \emph{Windowed Good--Turing} (\textsc{WingIt}) and show that its risk decays as $\widetilde{\mathcal{O}}(\mathsf{T_{mix}}/n)$, where $\mathsf{T_{mix}}$ denotes the mixing time of the chain in total variation distance. Notably, this rate is independent of the size of the state space and minimax-optimal up to a logarithmic factor in $n / \mathsf{T_{mix}}$. We also present a bound on the variance of the missing mass random variable, which may be of independent interest. We extend our estimator to approximate the stationary mass placed on elements occurring with small frequency in $X^n$. Finally, we demonstrate the efficacy of our estimators both in simulations on canonical chains and on sequences constructed from a popular natural language corpus., Comment: 40 pages, 5 figures

Classification models are expected to perform equally well for different classes, yet in practice, there are often large gaps in their performance. This issue of class bias is widely studied in cases of datasets with sample imbalance, but is relatively overlooked in balanced datasets. In this work, we introduce the concept of spectral imbalance in features as a potential source for class disparities and study the connections between spectral imbalance and class bias in both theory and practice. To build the connection between spectral imbalance and class gap, we develop a theoretical framework for studying class disparities and derive exact expressions for the per-class error in a high-dimensional mixture model setting. We then study this phenomenon in 11 different state-of-the-art pretrained encoders and show how our proposed framework can be used to compare the quality of encoders, as well as evaluate and combine data augmentation strategies to mitigate the issue. Our work sheds light on the class-dependent effects of learning, and provides new insights into how state-of-the-art pretrained features may have unknown biases that can be diagnosed through their spectra., Comment: 25 pages, 9 figures

We study the quantum nature of current through plasma-probe junction from the viewpoint of the metal probe. The intrinsic material properties of the metal and their influence on the nature of the observed current are theoretically worked out. The novel idea is that the plasma-sheath at the plasma-probe junction is treated as a potential barrier, and in analogy with the current conduction through a metal-metal junction, the current through the plasma-sheath is treated as a quantum barrier penetration problem. Essentially, we obtain an expression for the electron-current as a function of the bias voltage in its full range, thereby unlocking the intricate dependency of the current on the material properties of the probe., Comment: 5 pages, 5 figures

A mobile manipulator often finds itself in an application where it needs to take a close-up view before performing a manipulation task. Named this as a coupled active perception and manipulation (CAPM) problem, we model the uncertainty in the perception process and devise a key state/task planning approach that considers reachability conditions as task constraints of both perception and manipulation tasks for the mobile platform. By minimizing the expected energy usage in the body key state planning while satisfying task constraints, our algorithm achieves the best balance between the task success rate and energy usage. We have implemented the algorithm and tested it in both simulation and physical experiments. The results have confirmed that our algorithm has a lower energy consumption compared to a two-stage decoupled approach, while still maintaining a success rate of 100\% for the task., Comment: submitted to ICRA 2024

We present a novel approach to quantizing the length in noncommutative spaces with positional-dependent noncommutativity. The method involves constructing ladder operators that change the length not only along a plane but also along the third direction due to a noncommutative parameter that is a combination of canonical/Weyl-Moyal type and Lie algebraic type. The primary quantization of length in canonical-type noncommutative space takes place only on a plane, while in the present case, it happens in all three directions. We establish an operator algebra that allows for the raising or lowering of eigenvalues of the operator corresponding to the square of the length. We also attempt to determine how the obtained ladder operators act on different states and work out the eigenvalues of the square of the length operator in terms of eigenvalues corresponding to the ladder operators. We conclude by discussing the results obtained., Comment: 14 pages, 1 figure

Empirical risk minimization (ERM) of neural networks is prone to over-reliance on spurious correlations and poor generalization on minority groups. The recent deep feature reweighting (DFR) technique achieves state-of-the-art group robustness via simple last-layer retraining, but it requires held-out group and class annotations to construct a group-balanced reweighting dataset. In this work, we examine this impractical requirement and find that last-layer retraining can be surprisingly effective with no group annotations (other than for model selection) and only a handful of class annotations. We first show that last-layer retraining can greatly improve worst-group accuracy even when the reweighting dataset has only a small proportion of worst-group data. This implies a "free lunch" where holding out a subset of training data to retrain the last layer can substantially outperform ERM on the entire dataset with no additional data or annotations. To further improve group robustness, we introduce a lightweight method called selective last-layer finetuning (SELF), which constructs the reweighting dataset using misclassifications or disagreements. Our empirical and theoretical results present the first evidence that model disagreement upsamples worst-group data, enabling SELF to nearly match DFR on four well-established benchmarks across vision and language tasks with no group annotations and less than 3% of the held-out class annotations. Our code is available at https://github.com/tmlabonte/last-layer-retraining., Comment: NeurIPS 2023