Modern depth sensors can generate a huge number of 3D points in few seconds to be latter processed by Localization and Mapping algorithms. Ideally, these algorithms should handle efficiently large sizes of Point Clouds under the assumption that using more points implies more information available. The Eigen Factors (EF) is a new algorithm that solves SLAM by using planes as the main geometric primitive. To do so, EF exhaustively calculates the error of all points at complexity $O(1)$, thanks to the {\em Summation matrix} $S$ of homogeneous points. The solution of EF is highly efficient: i) the state variables are only the sensor poses -- trajectory, while the plane parameters are estimated previously in closed from and ii) EF alternating optimization uses a Newton-Raphson method by a direct analytical calculation of the gradient and the Hessian, which turns out to be a block diagonal matrix. Since we require to differentiate over eigenvalues and matrix elements, we have developed an intuitive methodology to calculate partial derivatives in the manifold of rigid body transformations $SE(3)$, which could be applied to unrelated problems that require analytical derivatives of certain complexity. We evaluate EF and other state-of-the-art plane SLAM back-end algorithms in a synthetic environment. The evaluation is extended to ICL dataset (RGBD) and LiDAR KITTI dataset. Code is publicly available at //github.com/prime-slam/EF-plane-SLAM.
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A major barrier to deploying current machine learning models lies in their non-reliability to dataset shifts. To resolve this problem, most existing studies attempted to transfer stable information to unseen environments. Particularly, independent causal mechanisms-based methods proposed to remove mutable causal mechanisms via the do-operator. Compared to previous methods, the obtained stable predictors are more effective in identifying stable information. However, a key question remains: which subset of this whole stable information should the model transfer, in order to achieve optimal generalization ability? To answer this question, we present a comprehensive minimax analysis from a causal perspective. Specifically, we first provide a graphical condition for the whole stable set to be optimal. When this condition fails, we surprisingly find with an example that this whole stable set, although can fully exploit stable information, is not the optimal one to transfer. To identify the optimal subset under this case, we propose to estimate the worst-case risk with a novel optimization scheme over the intervention functions on mutable causal mechanisms. We then propose an efficient algorithm to search for the subset with minimal worst-case risk, based on a newly defined equivalence relation between stable subsets. Compared to the exponential cost of exhaustively searching over all subsets, our searching strategy enjoys a polynomial complexity. The effectiveness and efficiency of our methods are demonstrated on synthetic data and the diagnosis of Alzheimer's disease.
The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O'Donnell (2017) and later Raghavendra & Weitz (2017) show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz (2017). In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.
It is well known that Cauchy problem for Laplace equations is an ill-posed problem in Hadamard's sense. Small deviations in Cauchy data may lead to large errors in the solutions. It is observed that if a bound is imposed on the solution, there exists a conditional stability estimate. This gives a reasonable way to construct stable algorithms. However, it is impossible to have good results at all points in the domain. Although numerical methods for Cauchy problems for Laplace equations have been widely studied for quite a long time, there are still some unclear points, for example, how to evaluate the numerical solutions, which means whether we can approximate the Cauchy data well and keep the bound of the solution, and at which points the numerical results are reliable? In this paper, we will prove the conditional stability estimate which is quantitatively related to harmonic measures. The harmonic measure can be used as an indicate function to pointwisely evaluate the numerical result, which further enables us to find a reliable subdomain where the local convergence rate is higher than a certain order.
Information borrowing from historical data is gaining attention in clinical trials of rare and pediatric diseases, where statistical power may be insufficient for confirmation of efficacy if the sample size is small. Although Bayesian information borrowing methods are well established, test-then-pool and equivalence-based test-then-pool methods have recently been proposed as frequentist methods to determine whether historical data should be used for statistical hypothesis testing. Depending on the results of the hypothesis testing, historical data may not be usable. This paper proposes a dynamic borrowing method for historical information based on the similarity between current and historical data. In our proposed method of dynamic information borrowing, as in Bayesian dynamic borrowing, the amount of borrowing ranges from 0% to 100%. We propose two methods using the density function of the t-distribution and a logistic function as a similarity measure. We evaluate the performance of the proposed methods through Monte Carlo simulations. We demonstrate the usefulness of borrowing information by reanalyzing actual clinical trial data.
Two journal-level indicators, respectively the mean ($m^i$) and the standard deviation ($v^i$) are proposed to be the core indicators of each journal and we show that quite several other indicators can be calculated from those two core indicators, assuming that yearly citation counts of papers in each journal follows more or less a log-normal distribution. Those other journal-level indicators include journal h index, journal one-by-one-sample comparison citation success index $S_j^i$, journal multiple-sample $K^i-K^j$ comparison success rate $S_{j,K^j}^{i,K^i }$, and minimum representative sizes $\kappa_j^i$ and $\kappa_i^j$, the average ranking of all papers in a journal in a set of journals($R^t$). We find that those indicators are consistent with those calculated directly using the raw citation data ($C^i=\{c_1^i,c_2^i,\dots,c_{N^i}^i \},\forall i$) of journals. In addition to its theoretical significance, the ability to estimate other indicators from core indicators has practical implications. This feature enables individuals who lack access to raw citation count data to utilize other indicators by simply using core indicators, which are typically easily accessible.
Training deep neural networks (DNNs) is computationally expensive, which is problematic especially when performing duplicated training runs, such as model ensemble or knowledge distillation. Once we have trained one DNN on some dataset, we have its learning trajectory (i.e., a sequence of intermediate parameters during training) which may potentially contain useful information for learning the dataset. However, there has been no attempt to utilize such information of a given learning trajectory for another training. In this paper, we formulate the problem of "transferring" a given learning trajectory from one initial parameter to another one, called learning transfer problem, and derive the first algorithm to approximately solve it by matching gradients successively along the trajectory via permutation symmetry. We empirically show that the transferred parameters achieve non-trivial accuracy before any direct training. Also, we analyze the loss landscape property of the transferred parameters, especially from a viewpoint of mode connectivity.
Evaluating simultaneous localization and mapping (SLAM) algorithms necessitates high-precision and dense ground truth (GT) trajectories. But obtaining desirable GT trajectories is sometimes challenging without GT tracking sensors. As an alternative, in this paper, we propose a novel prior-assisted SLAM system to generate a full six-degree-of-freedom ($6$-DOF) trajectory at around $10$Hz for benchmarking under the framework of the factor graph. Our degeneracy-aware map factor utilizes a prior point cloud map and LiDAR frame for point-to-plane optimization, simultaneously detecting degeneration cases to reduce drift and enhancing the consistency of pose estimation. Our system is seamlessly integrated with cutting-edge odometry via a loosely coupled scheme to generate high-rate and precise trajectories. Moreover, we propose a norm-constrained gravity factor for stationary cases, optimizing pose and gravity to boost performance. Extensive evaluations demonstrate our algorithm's superiority over existing SLAM or map-based methods in diverse scenarios in terms of precision, smoothness, and robustness. Our approach substantially advances reliable and accurate SLAM evaluation methods, fostering progress in robotics research.
We develop a new formulation of deep learning based on the Mori-Zwanzig (MZ) formalism of irreversible statistical mechanics. The new formulation is built upon the well-known duality between deep neural networks and discrete dynamical systems, and it allows us to directly propagate quantities of interest (conditional expectations and probability density functions) forward and backward through the network by means of exact linear operator equations. Such new equations can be used as a starting point to develop new effective parameterizations of deep neural networks, and provide a new framework to study deep-learning via operator theoretic methods. The proposed MZ formulation of deep learning naturally introduces a new concept, i.e., the memory of the neural network, which plays a fundamental role in low-dimensional modeling and parameterization. By using the theory of contraction mappings, we develop sufficient conditions for the memory of the neural network to decay with the number of layers. This allows us to rigorously transform deep networks into shallow ones, e.g., by reducing the number of neurons per layer (using projection operators), or by reducing the total number of layers (using the decay property of the memory operator).
For effective decision support in scenarios with conflicting objectives, sets of potentially optimal solutions can be presented to the decision maker. We explore both what policies these sets should contain and how such sets can be computed efficiently. With this in mind, we take a distributional approach and introduce a novel dominance criterion relating return distributions of policies directly. Based on this criterion, we present the distributional undominated set and show that it contains optimal policies otherwise ignored by the Pareto front. In addition, we propose the convex distributional undominated set and prove that it comprises all policies that maximise expected utility for multivariate risk-averse decision makers. We propose a novel algorithm to learn the distributional undominated set and further contribute pruning operators to reduce the set to the convex distributional undominated set. Through experiments, we demonstrate the feasibility and effectiveness of these methods, making this a valuable new approach for decision support in real-world problems.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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