Paraphrasing re-expresses meaning to enhance applications like text simplification, machine translation, and question-answering. Specific paraphrase types facilitate accurate semantic analysis and robust language models. However, existing paraphrase-type generation methods often misalign with human preferences due to reliance on automated metrics and limited human-annotated training data, obscuring crucial aspects of semantic fidelity and linguistic transformations. This study addresses this gap by leveraging a human-ranked paraphrase-type dataset and integrating Direct Preference Optimization (DPO) to align model outputs directly with human judgments. DPO-based training increases paraphrase-type generation accuracy by 3 percentage points over a supervised baseline and raises human preference ratings by 7 percentage points. A newly created human-annotated dataset supports more rigorous future evaluations. Additionally, a paraphrase-type detection model achieves F1 scores of 0.91 for addition/deletion, 0.78 for same polarity substitution, and 0.70 for punctuation changes. These findings demonstrate that preference data and DPO training produce more reliable, semantically accurate paraphrases, enabling downstream applications such as improved summarization and more robust question-answering. The PTD model surpasses automated metrics and provides a more reliable framework for evaluating paraphrase quality, advancing paraphrase-type research toward richer, user-aligned language generation and establishing a stronger foundation for future evaluations grounded in human-centric criteria.

Training-free guided generation is a widely used and powerful technique that allows the end user to exert further control over the generative process of flow/diffusion models. Generally speaking, two families of techniques have emerged for solving this problem for gradient-based guidance: namely, posterior guidance (i.e., guidance via projecting the current sample to the target distribution via the target prediction model) and end-to-end guidance (i.e., guidance by performing backpropagation throughout the entire ODE solve). In this work, we show that these two seemingly separate families can actually be unified by looking at posterior guidance as a greedy strategy of end-to-end guidance. We explore the theoretical connections between these two families and provide an in-depth theoretical of these two techniques relative to the continuous ideal gradients. Motivated by this analysis we then show a method for interpolating between these two families enabling a trade-off between compute and accuracy of the guidance gradients. We then validate this work on several inverse image problems and property-guided molecular generation.

The rising incidence of skin cancer, coupled with limited public awareness and a shortfall in clinical expertise, underscores an urgent need for advanced diagnostic aids. Artificial Intelligence (AI) has emerged as a promising tool in this domain, particularly for distinguishing malignant from benign skin lesions. Leveraging publicly available datasets of skin lesions, researchers have been developing AI-based diagnostic solutions. However, the integration of such computer systems in clinical settings is still nascent. This study aims to bridge this gap by employing a fusion of image processing techniques and machine learning algorithms, specifically neuro-fuzzy and colonial competition approaches. Applied to dermoscopic images from the ISIC database, our method achieved a notable accuracy of 94% on a dataset of 560 images. These results underscore the potential of our approach in aiding clinicians in the early detection of melanoma, thereby contributing significantly to skin cancer diagnostics.

This paper presents a differentiable optimization framework for the design of constrained Linear Differential Microphone Arrays (LDMAs). The proposed method leverages a non-uniform delay-and-sum beamformer as a light-weight base system model, proving its ability to achieve the optimal beampattern of LDMAs by jointly optimizing microphone positions and filter weights. The formulation enables the optimized design of a filter with a distortion-free constraint in the desired sound direction, while also imposing constraints on microphone positioning to ensure consistent performance. Through evaluation on multiple metrics, including Mean Squared Error (MSE), Directivity Index (DI), White Noise Gain (WNG), and computation time, and comparison with state-of-the-art methods, this approach demonstrates a flexible, directive, robust, and hardware-efficient design.

We derive universal approximation results for the class of (countably) $m$-rectifiable measures. Specifically, we prove that $m$-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on $[0,1]$ using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural networks required to achieve an approximation error of $\varepsilon$ is no larger than $2^{b(\varepsilon)}$ with $b(\varepsilon)=\mathcal{O}(\varepsilon^{-m}\log^2(\varepsilon))$. This result improves Lemma IX.4 in Perekrestenko et al. as it shows that the rate at which $b(\varepsilon)$ tends to infinity as $\varepsilon$ tends to zero equals the rectifiability parameter $m$, which can be much smaller than the ambient dimension. We extend this result to countably $m$-rectifiable measures and show that this rate still equals the rectifiability parameter $m$ provided that, among other technical assumptions, the measure decays exponentially on the individual components of the countably $m$-rectifiable support set.

Aggregation is a fundamental behavior for swarm robotics that requires a system to gather together in a compact, connected cluster. In 2014, Gauci et al. proposed a surprising algorithm that reliably achieves swarm aggregation using only a binary line-of-sight sensor and no arithmetic computation or persistent memory. It has been rigorously proven that this algorithm will aggregate one robot to another, but it remained open whether it would always aggregate a system of $n > 2$ robots as was observed in experiments and simulations. We prove that there exist deadlocked configurations from which this algorithm cannot achieve aggregation for $n > 3$ robots when the robots' motion is uniform and deterministic. On the positive side, we show that the algorithm (i) is robust to small amounts of error, enabling deadlock avoidance, and (ii) provably achieves a linear runtime speedup for the $n = 2$ case when using a cone-of-sight sensor. Finally, we introduce a noisy, discrete adaptation of this algorithm that is more amenable to rigorous analysis of noise and whose simulation results align qualitatively with the original, continuous algorithm.

In this paper orthogonal multifilters for astronomical image processing are presented. We obtained new orthogonal multifilters based on the orthogonal wavelet of Haar and Daubechies. Recently, multiwavelets have been introduced as a more powerful multiscale analysis tool. It adds several degrees of freedom in multifilter design and makes it possible to have several useful properties such as symmetry, orthogonality, short support, and a higher number of vanishing moments simultaneously. Multifilter decomposition of scanned photographic plates with astronomical images is made.

We propose a new method for the numerical solution of a PDE-driven model for colour image segmentation and give numerical examples of the results. The method combines the vector-valued Allen-Cahn phase field equation with initial data fitting terms. This method is known to be closely related to the Mumford-Shah problem and the level set segmentation by Chan and Vese. Our numerical solution is performed using a multigrid splitting of a finite element space, thereby producing an efficient and robust method for the segmentation of large images.

We propose distributed algorithms to automatically deploy a team of mobile robots to partition and provide coverage of a non-convex environment. To handle arbitrary non-convex environments, we represent them as graphs. Our partitioning and coverage algorithm requires only short-range, unreliable pairwise "gossip" communication. The algorithm has two components: (1) a motion protocol to ensure that neighboring robots communicate at least sporadically, and (2) a pairwise partitioning rule to update territory ownership when two robots communicate. By studying an appropriate dynamical system on the space of partitions of the graph vertices, we prove that territory ownership converges to a pairwise-optimal partition in finite time. This new equilibrium set represents improved performance over common Lloyd-type algorithms. Additionally, we detail how our algorithm scales well for large teams in large environments and how the computation can run in anytime with limited resources. Finally, we report on large-scale simulations in complex environments and hardware experiments using the Player/Stage robot control system.

The efficient realization of linear space-variant (non-convolution) filters is a challenging computational problem in image processing. In this paper, we demonstrate that it is possible to filter an image with a Gaussian-like elliptic window of varying size, elongation and orientation using a fixed number of computations per pixel. The associated algorithm, which is based on a family of smooth compactly supported piecewise polynomials, the radially-uniform box splines, is realized using pre-integration and local finite-differences. The radially-uniform box splines are constructed through the repeated convolution of a fixed number of box distributions, which have been suitably scaled and distributed radially in an uniform fashion. The attractive features of these box splines are their asymptotic behavior, their simple covariance structure, and their quasi-separability. They converge to Gaussians with the increase of their order, and are used to approximate anisotropic Gaussians of varying covariance simply by controlling the scales of the constituent box distributions. Based on the second feature, we develop a technique for continuously controlling the size, elongation and orientation of these Gaussian-like functions. Finally, the quasi-separable structure, along with a certain scaling property of box distributions, is used to efficiently realize the associated space-variant elliptical filtering, which requires O(1) computations per pixel irrespective of the shape and size of the filter.

Inspired by birds flying through cluttered environments such as dense forests, this paper studies the theoretical foundations of a novel motion planning problem: high-speed navigation through a randomly-generated obstacle field when only the statistics of the obstacle generating process are known a priori. Resembling a planar forest environment, the obstacle generating process is assumed to determine the locations and sizes of disk-shaped obstacles. When this process is ergodic, and under mild technical conditions on the dynamics of the bird, it is shown that the existence of an infinite collision-free trajectory through the forest exhibits a phase transition. On one hand, if the bird flies faster than a certain critical speed, then, with probability one, there is no infinite collision-free trajectory, i.e., the bird will eventually collide with some tree, almost surely, regardless of the planning algorithm governing the bird's motion. On the other hand, if the bird flies slower than this critical speed, then there exists at least one infinite collision-free trajectory, almost surely. Lower and upper bounds on the critical speed are derived for the special case of a homogeneous Poisson forest considering a simple model for the bird's dynamics. For the same case, an equivalent percolation model is provided. Using this model, the phase diagram is approximated in Monte-Carlo simulations. This paper also establishes novel connections between robot motion planning and statistical physics through ergodic theory and percolation theory, which may be of independent interest.

Unions of subspaces provide a powerful generalization to linear subspace models for collections of high-dimensional data. To learn a union of subspaces from a collection of data, sets of signals in the collection that belong to the same subspace must be identified in order to obtain accurate estimates of the subspace structures present in the data. Recently, sparse recovery methods have been shown to provide a provable and robust strategy for exact feature selection (EFS)--recovering subsets of points from the ensemble that live in the same subspace. In parallel with recent studies of EFS with L1-minimization, in this paper, we develop sufficient conditions for EFS with a greedy method for sparse signal recovery known as orthogonal matching pursuit (OMP). Following our analysis, we provide an empirical study of feature selection strategies for signals living on unions of subspaces and characterize the gap between sparse recovery methods and nearest neighbor (NN)-based approaches. In particular, we demonstrate that sparse recovery methods provide significant advantages over NN methods and the gap between the two approaches is particularly pronounced when the sampling of subspaces in the dataset is sparse. Our results suggest that OMP may be employed to reliably recover exact feature sets in a number of regimes where NN approaches fail to reveal the subspace membership of points in the ensemble.

We develop two approaches for analyzing the approximation error bound for the Nyström method, one based on the concentration inequality of integral operator, and one based on the compressive sensing theory. We show that the approximation error, measured in the spectral norm, can be improved from $O(N/\sqrt{m})$ to $O(N/m^{1 - ρ})$ in the case of large eigengap, where $N$ is the total number of data points, $m$ is the number of sampled data points, and $ρ\in (0, 1/2)$ is a positive constant that characterizes the eigengap. When the eigenvalues of the kernel matrix follow a $p$-power law, our analysis based on compressive sensing theory further improves the bound to $O(N/m^{p - 1})$ under an incoherence assumption, which explains why the Nyström method works well for kernel matrix with skewed eigenvalues. We present a kernel classification approach based on the Nyström method and derive its generalization performance using the improved bound. We show that when the eigenvalues of kernel matrix follow a $p$-power law, we can reduce the number of support vectors to $N^{2p/(p^2 - 1)}$, a number less than $N$ when $p > 1+\sqrt{2}$, without seriously sacrificing its generalization performance.

For the task of moving a group of indistinguishable agents on a connected graph with unit edge lengths into an arbitrary goal formation, it was previously shown that distance optimal paths can be scheduled to complete with a tight convergence time guarantee, using a fully centralized algorithm. In this study, we show that the problem formulation in fact induces a more fundamental ordering of the vertices on the underlying graph network, which directly leads to a more intuitive scheduling algorithm that assures the same convergence time and runs faster. More importantly, this structure enables a distributed scheduling algorithm once individual paths are assigned to the agents, which was not possible before. The vertex ordering also readily extends to more general graphs - those with non-unit capacities and edge lengths - for which we again guarantee the convergence time until the desired formation is achieved.

In this paper, we study the problem of optimal multi-robot path planning (MPP) on graphs. We propose two multiflow based integer linear programming (ILP) models that computes minimum last arrival time and minimum total distance solutions for our MPP formulation, respectively. The resulting algorithms from these ILP models are complete and guaranteed to yield true optimal solutions. In addition, our flexible framework can easily accommodate other variants of the MPP problem. Focusing on the time optimal algorithm, we evaluate its performance, both as a stand alone algorithm and as a generic heuristic for quickly solving large problem instances. Computational results confirm the effectiveness of our method.

This article presents near-optimal guarantees for accurate and robust image recovery from under-sampled noisy measurements using total variation minimization. In particular, we show that from O(slog(N)) nonadaptive linear measurements, an image can be reconstructed to within the best s-term approximation of its gradient up to a logarithmic factor, and this factor can be removed by taking slightly more measurements. Along the way, we prove a strengthened Sobolev inequality for functions lying in the null space of suitably incoherent matrices.

The problem of near-optimal distributed path planning to locally sensed targets is investigated in the context of large swarms. The proposed algorithm uses only information that can be locally queried, and rigorous theoretical results on convergence, robustness, scalability are established, and effect of system parameters such as the agent-level communication radius and agent velocities on global performance is analyzed. The fundamental philosophy of the proposed approach is to percolate local information across the swarm, enabling agents to indirectly access the global context. A gradient emerges, reflecting the performance of agents, computed in a distributed manner via local information exchange between neighboring agents. It is shown that to follow near-optimal routes to a target which can be only sensed locally, and whose location is not known a priori, the agents need to simply move towards its "best" neighbor, where the notion of "best" is obtained by computing the state-specific language measure of an underlying probabilistic finite state automata. The theoretical results are validated in high-fidelity simulation experiments, with excess of $10^4$ agents.

In a Role-Playing Game, finding optimal trajectories is one of the most important tasks. In fact, the strategy decision system becomes a key component of a game engine. Determining the way in which decisions are taken (online, batch or simulated) and the consumed resources in decision making (e.g. execution time, memory) will influence, in mayor degree, the game performance. When classical search algorithms such as A* can be used, they are the very first option. Nevertheless, such methods rely on precise and complete models of the search space, and there are many interesting scenarios where their application is not possible. Then, model free methods for sequential decision making under uncertainty are the best choice. In this paper, we propose a heuristic planning strategy to incorporate the ability of heuristic-search in path-finding into a Dyna agent. The proposed Dyna-H algorithm, as A* does, selects branches more likely to produce outcomes than other branches. Besides, it has the advantages of being a model-free online reinforcement learning algorithm. The proposal was evaluated against the one-step Q-Learning and Dyna-Q algorithms obtaining excellent experimental results: Dyna-H significantly overcomes both methods in all experiments. We suggest also, a functional analogy between the proposed sampling from worst trajectories heuristic and the role of dreams (e.g. nightmares) in human behavior.

Cyclic coordinate descent is a classic optimization method that has witnessed a resurgence of interest in machine learning. Reasons for this include its simplicity, speed and stability, as well as its competitive performance on $\ell_1$ regularized smooth optimization problems. Surprisingly, very little is known about its finite time convergence behavior on these problems. Most existing results either just prove convergence or provide asymptotic rates. We fill this gap in the literature by proving $O(1/k)$ convergence rates (where $k$ is the iteration counter) for two variants of cyclic coordinate descent under an isotonicity assumption. Our analysis proceeds by comparing the objective values attained by the two variants with each other, as well as with the gradient descent algorithm. We show that the iterates generated by the cyclic coordinate descent methods remain better than those of gradient descent uniformly over time.

We study the problem of distributed traffic control in the partitioned plane, where the movement of all entities (robots, vehicles, etc.) within each partition (cell) is coupled. Establishing liveness in such systems is challenging, but such analysis will be necessary to apply such distributed traffic control algorithms in applications like coordinating robot swarms and the intelligent highway system. We present a formal model of a distributed traffic control protocol that guarantees minimum separation between entities, even as some cells fail. Once new failures cease occurring, in the case of a single target, the protocol is guaranteed to self-stabilize and the entities with feasible paths to the target cell make progress towards it. For multiple targets, failures may cause deadlocks in the system, so we identify a class of non-deadlocking failures where all entities are able to make progress to their respective targets. The algorithm relies on two general principles: temporary blocking for maintenance of safety and local geographical routing for guaranteeing progress. Our assertional proofs may serve as a template for the analysis of other distributed traffic control protocols. We present simulation results that provide estimates of throughput as a function of entity velocity, safety separation, single-target path complexity, failure-recovery rates, and multi-target path complexity.

We present a fast convolution-based technique for computing an approximate, signed Euclidean distance function $S$ on a set of 2D and 3D grid locations. Instead of solving the non-linear, static Hamilton-Jacobi equation ($\|\nabla S\|=1$), our solution stems from first solving for a scalar field $ϕ$ in a linear differential equation and then deriving the solution for $S$ by taking the negative logarithm. In other words, when $S$ and $ϕ$ are related by $ϕ= \exp \left(-\frac{S}τ \right)$ and $ϕ$ satisfies a specific linear differential equation corresponding to the extremum of a variational problem, we obtain the approximate Euclidean distance function $S = -τ\log(ϕ)$ which converges to the true solution in the limit as $τ\rightarrow 0$. This is in sharp contrast to techniques like the fast marching and fast sweeping methods which directly solve the Hamilton-Jacobi equation by the Godunov upwind discretization scheme. Our linear formulation results in a closed-form solution to the approximate Euclidean distance function expressible as a discrete convolution, and hence efficiently computable using the fast Fourier transform (FFT). Our solution also circumvents the need for spatial discretization of the derivative operator. As $τ\rightarrow0$ we show the convergence of our results to the true solution and also bound the error for a given value of $τ$. The differentiability of our solution allows us to compute---using a set of convolutions---the first and second derivatives of the approximate distance function. In order to determine the sign of the distance function (defined to be positive inside a closed region and negative outside), we compute the winding number in 2D and the topological degree in 3D, whose computations can also be performed via fast convolutions. We demonstrate the efficacy of our method through a set of experimental results.

A number of representation schemes have been presented for use within learning classifier systems, ranging from binary encodings to neural networks. This paper presents results from an investigation into using discrete and fuzzy dynamical system representations within the XCSF learning classifier system. In particular, asynchronous random Boolean networks are used to represent the traditional condition-action production system rules in the discrete case and asynchronous fuzzy logic networks in the continuous-valued case. It is shown possible to use self-adaptive, open-ended evolution to design an ensemble of such dynamical systems within XCSF to solve a number of well-known test problems.

This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which exploits a certain tensor structure in their low-order observable moments (typically, of second- and third-order). Specifically, parameter estimation is reduced to the problem of extracting a certain (orthogonal) decomposition of a symmetric tensor derived from the moments; this decomposition can be viewed as a natural generalization of the singular value decomposition for matrices. Although tensor decompositions are generally intractable to compute, the decomposition of these specially structured tensors can be efficiently obtained by a variety of approaches, including power iterations and maximization approaches (similar to the case of matrices). A detailed analysis of a robust tensor power method is provided, establishing an analogue of Wedin's perturbation theorem for the singular vectors of matrices. This implies a robust and computationally tractable estimation approach for several popular latent variable models.

A number of representation schemes have been presented for use within Learning Classifier Systems, ranging from binary encodings to neural networks. This paper presents results from an investigation into using a discrete dynamical system representation within the XCS Learning Classifier System. In particular, asynchronous random Boolean networks are used to represent the traditional condition-action production system rules. It is shown possible to use self-adaptive, open-ended evolution to design an ensemble of such discrete dynamical systems within XCS to solve a number of well-known test problems.

We present a novel algebraic combinatorial view on low-rank matrix completion based on studying relations between a few entries with tools from algebraic geometry and matroid theory. The intrinsic locality of the approach allows for the treatment of single entries in a closed theoretical and practical framework. More specifically, apart from introducing an algebraic combinatorial theory of low-rank matrix completion, we present probability-one algorithms to decide whether a particular entry of the matrix can be completed. We also describe methods to complete that entry from a few others, and to estimate the error which is incurred by any method completing that entry. Furthermore, we show how known results on matrix completion and their sampling assumptions can be related to our new perspective and interpreted in terms of a completability phase transition.

A possibly immortal agent tries to maximise its summed discounted rewards over time, where discounting is used to avoid infinite utilities and encourage the agent to value current rewards more than future ones. Some commonly used discount functions lead to time-inconsistent behavior where the agent changes its plan over time. These inconsistencies can lead to very poor behavior. We generalise the usual discounted utility model to one where the discount function changes with the age of the agent. We then give a simple characterisation of time-(in)consistent discount functions and show the existence of a rational policy for an agent that knows its discount function is time-inconsistent.

We evaluate natural gradient, an algorithm originally proposed in Amari (1997), for learning deep models. The contributions of this paper are as follows. We show the connection between natural gradient and three other recently proposed methods for training deep models: Hessian-Free (Martens, 2010), Krylov Subspace Descent (Vinyals and Povey, 2012) and TONGA (Le Roux et al., 2008). We describe how one can use unlabeled data to improve the generalization error obtained by natural gradient and empirically evaluate the robustness of the algorithm to the ordering of the training set compared to stochastic gradient descent. Finally we extend natural gradient to incorporate second order information alongside the manifold information and provide a benchmark of the new algorithm using a truncated Newton approach for inverting the metric matrix instead of using a diagonal approximation of it.

We present a method to track the precise shape of an object in video based on new modeling and optimization on a new Riemannian manifold of parameterized regions. Joint dynamic shape and appearance models, in which a template of the object is propagated to match the object shape and radiance in the next frame, are advantageous over methods employing global image statistics in cases of complex object radiance and cluttered background. In cases of 3D object motion and viewpoint change, self-occlusions and dis-occlusions of the object are prominent, and current methods employing joint shape and appearance models are unable to adapt to new shape and appearance information, leading to inaccurate shape detection. In this work, we model self-occlusions and dis-occlusions in a joint shape and appearance tracking framework. Self-occlusions and the warp to propagate the template are coupled, thus a joint problem is formulated. We derive a coarse-to-fine optimization scheme, advantageous in object tracking, that initially perturbs the template by coarse perturbations before transitioning to finer-scale perturbations, traversing all scales, seamlessly and automatically. The scheme is a gradient descent on a novel infinite-dimensional Riemannian manifold that we introduce. The manifold consists of planar parameterized regions, and the metric that we introduce is a novel Sobolev-type metric defined on infinitesimal vector fields on regions. The metric has the property of resulting in a gradient descent that automatically favors coarse-scale deformations (when they reduce the energy) before moving to finer-scale deformations. Experiments on video exhibiting occlusion/dis-occlusion, complex radiance and background show that occlusion/dis-occlusion modeling leads to superior shape accuracy compared to recent methods employing joint shape/appearance models or employing global statistics.

In many signal processing applications, one wishes to acquire images that are sparse in transform domains such as spatial finite differences or wavelets using frequency domain samples. For such applications, overwhelming empirical evidence suggests that superior image reconstruction can be obtained through variable density sampling strategies that concentrate on lower frequencies. The wavelet and Fourier transform domains are not incoherent because low-order wavelets and low-order frequencies are correlated, so compressive sensing theory does not immediately imply sampling strategies and reconstruction guarantees. In this paper we turn to a more refined notion of coherence -- the so-called local coherence -- measuring for each sensing vector separately how correlated it is to the sparsity basis. For Fourier measurements and Haar wavelet sparsity, the local coherence can be controlled and bounded explicitly, so for matrices comprised of frequencies sampled from a suitable inverse square power-law density, we can prove the restricted isometry property with near-optimal embedding dimensions. Consequently, the variable-density sampling strategy we provide allows for image reconstructions that are stable to sparsity defects and robust to measurement noise. Our results cover both reconstruction by $\ell_1$-minimization and by total variation minimization. The local coherence framework developed in this paper should be of independent interest in sparse recovery problems more generally, as it implies that for optimal sparse recovery results, it suffices to have bounded \emph{average} coherence from sensing basis to sparsity basis -- as opposed to bounded maximal coherence -- as long as the sampling strategy is adapted accordingly.

The CUR matrix decomposition and the Nyström approximation are two important low-rank matrix approximation techniques. The Nyström method approximates a symmetric positive semidefinite matrix in terms of a small number of its columns, while CUR approximates an arbitrary data matrix by a small number of its columns and rows. Thus, CUR decomposition can be regarded as an extension of the Nyström approximation. In this paper we establish a more general error bound for the adaptive column/row sampling algorithm, based on which we propose more accurate CUR and Nyström algorithms with expected relative-error bounds. The proposed CUR and Nyström algorithms also have low time complexity and can avoid maintaining the whole data matrix in RAM. In addition, we give theoretical analysis for the lower error bounds of the standard Nyström method and the ensemble Nyström method. The main theoretical results established in this paper are novel, and our analysis makes no special assumption on the data matrices.