Supervised machine learning (ML) methods are emerging as valid alternatives to standard mathematical methods for identifying knots in long, collapsed polymers. Here, we introduce a hybrid supervised/unsupervised ML approach for knot classification based on a variational autoencoder enhanced with a knot type classifier (VAEC). The neat organization of knots in its latent representation suggests that the VAEC, only based on an arbitrary labeling of three-dimensional configurations, has grasped complex topological concepts such as chirality, unknotting number, braid index, and the grouping in families such as achiral, torus, and twist knots. The understanding of topological concepts is confirmed by the ability of the VAEC to distinguish the chirality of knots $9_{42}$ and $10_{71}$ not used for its training and with a notoriously undetected chirality to standard tools. The well-organized latent space is also key for generating configurations with the decoder that reliably preserves the topology of the input ones. Our findings demonstrate the ability of a hybrid supervised-generative ML algorithm to capture different topological features of entangled filaments and to exploit this knowledge to faithfully reconstruct or produce new knotted configurations without simulations.

We prove that the detection rate of n-crossing alternating links by many standard link invariants decays exponentially in n, implying that they detect alternating links with probability zero. This phenomenon applies broadly, in particular to the Jones and HOMFLYPT polynomials and integral Khovanov homology. We also use a big-data approach to analyze knots and provide evidence that, for knots as well, these invariants exhibit the same asymptotic failure of detection.

SplineNets are continuous generalizations of neural decision graphs, and they can dramatically reduce runtime complexity and computation costs of CNNs, while maintaining or even increasing accuracy. Functions of SplineNets are both dynamic ( i. e., conditioned on the input) and hierarchical ( i .e ., conditioned on the computational path). SplineNets employ a unified loss function with a desired level of smoothness over both the network and decision parameters, while allowing for sparse activation of a subset of nodes for individual samples. In particular, we embed infinitely many function weights (e. g. filters) on smooth, low dimensional manifolds parameterized by compact B-splines, which are indexed by a position parameter. Instead of sampling from a categorical distribution to pick a branch, samples choose a continuous position to pick a function weight. We further show that by maximizing the mutual information between spline positions and class labels, the network can be optimally utilized and specialized for classification tasks. Experiments show that our approach can significantly increase the accuracy of ResNets with negligible cost in speed, matching the precision of a 110 level ResNet with a 32 level SplineNet.

Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods.

Abstract--Autonomous robotic systems heavily rely on environment knowledge to safely navigate. For search & rescue, a flying robot requires robust real-time perception, enabled by complementary sensors. IMU data constrains acceleration and rotation, whereas LiDAR measures accurate distances around the robot. Our new scan window uses non-uniform temporal knot placement to ensure continuity over the whole trajectory without additional scan delay. Moreover, we accelerate essential covariance and GMM computations with Kronecker sums and products by a factor of 3.3. An unscented transform de-skews surfels, while a splitting into intra-scan segments facilitates motion compensation during spline optimization. Complementary soft constraints on relative poses and preintegrated IMU pseudo-measurements further improve robustness and accuracy. ELIABLE real-time perception is essential for robotic autonomy. In particular, accurate mapping and ego-motion estimation are key components for safe interaction in complex and unstructured environments. Due to their precision and measurement density, modern LiDARs are often used in these scenarios, e.g., in the DARP A Subterranean Challenge [1], [2]. Sensor motion during scanning distorts the point cloud and degrades the quality of the map. This intra-scan motion is either compensated by de-skewing prior to registration [3], [4], [5], [6] or by modeling it with a continuous-time trajectory [7], [8], [9]. The former uses the previous state estimate and, optionally, an IMU to predict the motion and transform points to a common reference time. However, this comes at the cost of reduced real-time capability and requires either costly reintegration of surfels [9] or a limited number of selected pointwise features [e.g., CT -ICP [7], CLINS [8]]. To overcome these limitations of continuous-time methods, our novel real-time LiDAR-inertial odometry (LIO) jointly optimizes temporally partitioned scan segments (Figure 1) by registering multi-resolution surfel maps while an unscented transform (UT) compensates the intra-surfel motion. Manuscript received October XX, 2025; revised XX, 2025.

SplineNets are continuous generalizations of neural decision graphs, and they can dramatically reduce runtime complexity and computation costs of CNNs, while maintaining or even increasing accuracy. Functions of SplineNets are both dynamic ( i. e., conditioned on the input) and hierarchical ( i .e ., conditioned on the computational path). SplineNets employ a unified loss function with a desired level of smoothness over both the network and decision parameters, while allowing for sparse activation of a subset of nodes for individual samples. In particular, we embed infinitely many function weights (e. g. filters) on smooth, low dimensional manifolds parameterized by compact B-splines, which are indexed by a position parameter. Instead of sampling from a categorical distribution to pick a branch, samples choose a continuous position to pick a function weight. We further show that by maximizing the mutual information between spline positions and class labels, the network can be optimally utilized and specialized for classification tasks. Experiments show that our approach can significantly increase the accuracy of ResNets with negligible cost in speed, matching the precision of a 110 level ResNet with a 32 level SplineNet.

Detecting changepoints in a one-dimensional signal is a classical yet fundamental problem. The fused lasso provides an elegant convex formulation that produces a stepwise estimate of the mean, but quantifying the uncertainty of the detected changepoints remains difficult. Post-selection inference (PSI) offers a principled way to compute valid $p$-values after a data-driven selection, but its application to the fused lasso has been considered computationally cumbersome, requiring the tracking of many ``hit'' and ``leave'' events along the regularization path. In this paper, we show that the one-dimensional fused lasso has a surprisingly simple geometry: each changepoint enters in a strictly one-sided fashion, and there are no leave events. This structure implies that the so-called \emph{conservative spacing test} of Tibshirani et al.\ (2016), previously regarded as an approximation, is in fact \emph{exact}. The truncation region in the selective law reduces to a single lower bound given by the next knot on the LARS path. As a result, the exact selective $p$-value takes a closed form identical to the simple spacing statistic used in the LARS/lasso setting, with no additional computation. This finding establishes one of the rare cases in which an exact PSI procedure for the generalized lasso admits a closed-form pivot. We further validate the result by simulations and real data, confirming both exact calibration and high power. Keywords: fused lasso; changepoint detection; post-selection inference; spacing test; monotone LASSO

Neural networks are parametric and powerful tools for function approximation, and the choice of architecture heavily influences their interpretability, efficiency, and generalization. In contrast, Gaussian processes (GPs) are nonparametric probabilistic models that define distributions over functions using a kernel to capture correlations among data points. However, these models become computationally expensive for large-scale problems, as they require inverting a large covariance matrix. Kolmogorov- Arnold networks (KANs), semi-parametric neural architectures, have emerged as a prominent approach for modeling complex functions with structured and efficient representations through spline layers. Kurkova Kolmogorov-Arnold networks (KKANs) extend this idea by reducing the number of spline layers in KAN and replacing them with Chebyshev layers and multi-layer perceptrons, thereby mapping inputs into higher-dimensional spaces before applying spline-based transformations. Compared to KANs, KKANs perform more stable convergence during training, making them a strong architecture for estimating operators and system modeling in dynamical systems. By enhancing the KKAN architecture, we develop a novel learning algorithm, distance-aware error for Kurkova-Kolmogorov networks (K-DAREK), for efficient and interpretable function approximation with uncertainty quantification. Our approach establishes robust error bounds that are distance-aware; this means they reflect the proximity of a test point to its nearest training points. Through case studies on a safe control task, we demonstrate that K-DAREK is about four times faster and ten times higher computationally efficiency than Ensemble of KANs, 8.6 times more scalable than GP by increasing the data size, and 50% safer than our previous work distance-aware error for Kolmogorov networks (DAREK).

Remarkably, the solutions learned for each individual task resemble those obtained by solving a kernel regression problem, revealing a novel connection between neural networks and kernel methods.