sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: When the cardinality of X S increases under infill asymptotics [14, §3.3], This is the limit for the posterior variance at any test location for task T , if one has training data only for the secondary task S. This is because a correlation of ρ between the tasks prevents any training location for task S from having correlation higher than ρ with a test location for task T . Suppose correlations in the input-space are given by an isotropic covariance function k x (|x − x |).
Sentence 2: if we translate correlations into distances between data locations, then any training location from task S is beyond a certain radius from any test location for task T . a training location from task T may lay arbitrarily close to a test location for task T , subject to the constraints of noise.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We denote the weights of this linear combination as W 2 < NVOX⇥NCELLS , and u t as a vector of size N VOX representing predicted neural activity at each voxel at time t. Thus, For training the model, the predicted neural activity is compared with the actual activity recorded by the scanner.
Sentence 2: neural activity is not instantly reflected in the intensity recorded by the scanner, but is delayed according to the haemodynamic response function (HRF; Figure S3).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Moreover, among those few exceptions that do not use projections onto X i when Π Xi is not easy to compute, only [15,16] can handle agent-specific constraints without assuming global knowledge of the constraints by all agents.
Sentence 2: no rate results in terms of suboptimality, local infeasibility, and consensus violation exist for the primaldual distributed methods in [15,16] when implemented for the agent-specific conic constraint sets X i = {x : A i x − b i ∈ K i } studied in this paper.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If this is the case then PickyAdaBoost abstains in that round and does not include h t into the combined hypothesis it is constructing.
Sentence 2: (Note that consequently the distribution for the next round of boosting will also be D .) if the current base classifier has advantage γ where |γ| ≥ γ, then PickyAd-aBoost proceeds to use the weak hypothesis just like AdaBoost, i.e.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The cost function in equation ( 5) is the square of the Frobenius norm of the difference between the empirical matrix ¢ and the fit kernel ¡ c . The use of the Frobenius norm is similar to the Ordinary Least Squares technique of fitting variogram parameters in geostatistics [7].
Sentence 2: instead of summing variogram estimates within spatial bins, we form covariance estimates over all meta-training data pairs ¥ .
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, in the case (d = 3, k = 1) the free parameters are Z12 and Z13, which define a coordinate system for the sphere.
Sentence 2: as a function of U, the integrand is simply 1 The density is maximized when U contains the top k eigenvectors of S . the density is unchanged if we negate any column of U.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For real datasets, such as image histograms, where minwise sampling is popular [13], the value of this sparsity is of the order of 0.02-0.08 (see Section 4.2) leading to 1 s x ≈ 13 − 50.
Sentence 2: the number of non-zeros is around half million.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When from the observations, a red node w ∈ V (r) is connected to at most a single green node, i.e., if v∈V (g) A vw ≤ 1, this red node is useless in the classification of green nodes.
Sentence 2: when a red node is connected to two green nodes, say v 1 and v 2 (A v1w = 1 = A v2w ), we may infer that the green nodes v 1 and v 2 are likely to be in the same cluster.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Control was historically among the earliest applications of neural networks, but the recent surge in performance has been in computer vision, speech recognition and other classification problems that arise in artificial intelligence and machine learning, where large datasets are available.
Sentence 2: the data needed to learn neural network controllers is much harder to obtain, and in the case of imaginary characters and novel robots we have to synthesize the training data ourselves (via trajectory optimization).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The spike counts are conditionally Poisson distributed given a vector of parameters w and time-dependent vector of covariates x t The log-likelihood of w given the vector of all observed spike counts y is where we have dropped terms constant in w and the t-th row of the design matrix X is x t . The methods in this paper apply both when using the canonical log link function such that the nonlinearity is f (x) = exp(x) and when using alternative nonlinearities.
Sentence 2: with moderate amounts of data, first or second order optimization techniques can be used to quickly find point estimates of the parameters w. inference can be prohibitive for large datasets, as each evaluation of the log-likelihood requires passing through the entire design matrix.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: For example, any F . With this scenario in mind, it is unsurprising that low-rank approximation guarantees fail as an accuracy measure in practice.
Sentence 2: we ran a standard sketch-and-solve approximate SVD algorithm (see Section 3) on SNAP/AMAZON0302, an Amazon product co-purchasing dataset [22,23], and achieved very good low-rank approximation error in both norms for k = 30: the approximate principal components given by Z are of significantly lower quality than A's true singular vectors (see Figure 1).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Table 1 shows that for the first three sets of data (Gaussians, Circles, AI) maximum margin and spectral clustering obtained identical small error rates, which were in turn significantly smaller than those obtained by k-means.
Sentence 2: maximum margin clustering demonstrates a substantial advantage on the fourth data set (Joined Circles) over both spectral and k-means clustering.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Summary In our framework, SGLD and SGRLD take Q(z) = 0 and instead stress the design of the diffusion matrix D(z), with SGLD using a constant D(z) and SGRLD an adaptive, θ-dependent diffusion matrix to better account for the geometry of the space being explored.
Sentence 2: hMC takes D(z) = 0 and focuses on the curl matrix Q(z).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: From high level, our proof constructs an estimation sequence { w (t) , M (t) , t } such that t → 0 and w * − w (t) 2 + M * − M (t) 2 ≤ t . In conventional matrix sensing, this construction is possible when the sensing matrix satisfies the Restricted Isometric Property (RIP) [Candès and Recht, 2009]: When A is 2 -norm δ k -RIP for any rank k matrix M , A A is nearly isometric [Jain et al., 2012], which implies M − A A(M )/n 2 ≤ δ.
Sentence 2: then we can construct our estimation sequence as following: in gFM and symmetric rank-one matrix sensing, the 2 -norm RIP condition cannot be satisfied with high probability [Cai and Zhang, 2015].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Finally, we exemplified our work by using KD-trees as the tree-consistent partition structure for generating the component-specific partitions in CS-EM, which limited its effectiveness in high dimensions.
Sentence 2: any hierarchical partition structure can be used, and the work in [8] therefore suggest that changing to an anchor tree (a special kind of metric tree [15]) will also render CS-EM effective in high dimensions, under the assumption of lower intrinsic dimensionality for the data.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In the Gaussian setting, the minimum sample complexity can be improved to n = Ω(∆ 2 log p), i.e., when J min = Θ(1/ √ ∆) where the maximum degree scales as ∆ = Θ(log p log c) [7].
Sentence 2: for Ising models, the minimum sample complexity can be further improved to n = Ω(c 4 log p), i.e., when J min = Θ(J * ) = Θ(1/c).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Economists and computer scientists are often concerned with inferring people's preferences from their choices, developing econometric methods (e.g., [1,2]) and collaborative filtering algorithms (e.g., [3,4,5]) that will allow them to assess the subjective value of an item or determine which other items a person might like.
Sentence 2: identifying the preferences of others is also a key part of social cognitive development, allowing children to understand how people act and what they want.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: We now define the expected reward to be the adaptive value of information of extracting the a'th set of features given the system state and budget B: Intuitively, (3) says that each time we add additional features to the computation, we gain reward equal to the decrease in error achieved with the new features (or pay a penalty if the error increases.)
Sentence 2: if we ever exceed the budget, then any further decrease does not count; no more reward can be gained.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Although fMRI is the most popular method for functional brain imaging with high spatial resolution, it suffers from poor temporal resolution since it measures blood oxygenation level signals with fluctuations in the order of seconds.
Sentence 2: dynamic neuronal activity has fluctuations in the sub-millisecond time-scale that can only be directly measured with electromagnetic source imaging (ESI).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: These developments have made it possible to apply RNNs to new domains such as language translation [1,40] and parsing [44], and image and video captioning [7,45].
Sentence 2: the current RNNs are designed to output each time one "token" in the input sequence, they can not handle properly the segment detection task in which each time a continuous chunk of the inputs is selected.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 1BitSGD was experimentally observed to preserve convergence [35], under certain conditions; thanks to the reduction in communication, it enabled state-of-the-art scaling of deep neural networks (DNNs) for acoustic modelling [37].
Sentence 2: it is currently not known if 1BitSGD provides any guarantees, even under strong assumptions, and it is not clear if higher compression is achievable.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To extract convolutional features for source D s and target D t , the input source and target images (I s , I t ) are first passed through fully-convolutional feature extraction networks with shared parameters W F such that D i = F(I i |W F ), and the feature for each pixel then undergoes L 2 normalization.
Sentence 2: in the recurrent formulation, at each iteration the target features D t can be extracted according to extracting each feature by transforming local receptive fields within the target image I t according to T i for each pixel i and then passing it through the networks would be time-consuming when iterating the networks.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: A sufficiently weak dependence results in a monotonically increasing Bayes factor which favors the absence of the edge A +-B at any finite value of 0:.
Sentence 2: given a sufficiently strong dependence between A and B, the log Bayes factor takes on positive values for all (finite) 0: exceeding a certain value 0:+ of the scale parameter.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: These estimators enjoy the parametric rate of O n −1/2 when β > d/4, and work by optimally estimating the density and then applying a correction to the plug-in estimate.
Sentence 2: our estimator undersmooths the density, and converges at a slower rate of O n −β/(β+d) when β < d (and the parametric rate O n −1/2 when β ≥ d), but obeys an exponential concentration inequality, which is not known for the estimators of [8].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: In practice, their model is defined by the same forward equations as ours.
Sentence 2: equation 3 which computes the backward vectors is instead: Incorrect "@Hannah Sunder: The Walking Dead is just a great tv show" its bad ass just started to watch the 2nd season to catch up with the 3rd Figure 4: Examples of predictions made by the GB-RNN for twitter documents.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Empirical simulations showing that the other forward algorithms also suffer this regret are in Appendix F. An attractive feature of forward algorithms is that they generalize to partial orders, for which efficient offline optimization algorithms exist.
Sentence 2: in Section 4 we saw that FAs only give a Õ(t − 1 2 ) rate, while in Section 3 we saw that Õ(t − 2 3 ) is possible (with an algorithm that is not known to scale to partial orders).
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Furthermore, when some of the singular values of M fall below the "noise level" √ dσ, one can show a tighter bound, with a nearly-optimal bias-variance tradeoff; see Theorem 2.7 in [5] for details.
Sentence 2: when M is full-rank, then the error of M depends on the behavior of the tail M c . We will consider a couple of cases.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When µ 1 = 1, µ 2 = 0, the attacker is interested in increasing the RMSE of the collaborative filtering system and hence reducing the system's availability.
Sentence 2: when µ 1 = 1, µ 2 = −1 the attacker wishes to increase RMSE while at the same time keeping the rating of specific items (j 0 ) as low as possible for certain malicious purposes.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Unfortunately, both variants suffer from a polynomial time complexity with a super-linear dependence on the dimensionality d (at least a power of 4), which renders them not practical for optimizing problems of high dimension.
Sentence 2: second-order information carried by the Hessian has been utilized to escape from a saddle point, which usually yields an almost linear time complexity in terms of the dimensionality d under the assumption that the Hessian-vector product (HVP) can be performed in a linear time.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Thus, according to these schemes, patterns are stored wholesale in the hippocampus when they first appear, and are continually read back to cortex to cause plasticity along with the new information.
Sentence 2: if the hippocampus is permanently required to prevent a catastrophe, then, first, there is no true consolidation: if neocortical plasticity is not inhibited by hippocampal damage, 20 then its integrity is permanently required to prevent degradation; and, second, what is the point of consolidation -couldn't the hippocampus suffice by itself?
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Existing approaches to clustering HMMs operate directly on the HMM parameter space, by grouping HMMs according to a suitable pairwise distance defined in terms of the HMM parameters.
Sentence 2: as HMM parameters lie on a non-linear manifold, a simple application of the k-means algorithm will not succeed in the task, since it assumes real vectors in a Euclidean space.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: All units and parameters at all levels of the network are engaged in representing any given input and are adjusted together during learning.
Sentence 2: we argue that one-shot learning of new classes will be easier in architectures that can explicitly identify only a small number of degrees of freedom (latent variables and parameters) that are relevant to the new concept being learned, and thereby achieve more appropriate and flexible transfer of learned representations to new tasks.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Like Nash equilibrium, a QPE and EFPE can be computed in polynomial time in the size of the input game.
Sentence 2: the big-O complexity hides dramatically larger constants in the case of a QPE or an EFPE, and the algorithms known so far thus do not scale beyond small instances [ Čermák et al., 2014;Ganzfried and Sandholm, 2015].
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If the ith sensor is not involved in a reward communication event at that time, its global reward estimate is updated according to Y i (k) = αY i (k − 1) + r i (k).
Sentence 2: at any time k that there is a communication event, its global reward estimate is updated according to , where j is the index of the sensor with which communication occurs.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: This demonstrates that naively learning from synthetic images can be problematic due to a gap between synthetic and real image distributions -synthetic data is often not realistic enough with artifacts and severe texture losses, misleading the network to overfit to fake information only presented in synthetic images and fail to generalize well on real data.
Sentence 2: with the injection of photorealistic and identity preserving faces generated by DA-GAN without extra human annotation efforts, our method outperforms b1 by 1.00% for TAR @ FAR=0.001 of verification and 1.50% for FNIR @ FPIR=0.01, 0.50% for Rank-1 of identification.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Since in LCC each data is encoded by some anchor points on the data manifold, it can model the decision boundary of an SVM directly using f (x) ≈ v∈C γ v (x)f (v).
Sentence 2: then by taking γ x as the input data of a linear SVM, f (v)'s can be learned to approximate the decision boundary f . OCC learns a set of orthogonal basis vectors, rather than anchor points, and corresponding coding for data.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Specifically, because of the assumed independence of the {v i }, the EM method requires one to repeatedly maximize the Q-function such that the estimate of α at the (m + 1)th iteration is: Like the compound Dirichlet likelihood, the compound shadow Dirichlet likelihood is not necessarily concave.
Sentence 2: note that the Q-function given in ( 7) is concave, because log p(v i |α) = − log |det(M )| + log p D,α M −1 v i , where p D,α is the Dirichlet distribution with parameter α, and by a theorem of Ronning [11], log p D,α is a concave function, and adding a constant does not change the concavity.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 1, the activation of the last layer is therefore equal to where we defined the total weight matrix product W in the last expression, equal to the chain of matrix multiplications along all layers 1, 2, . . . , L. This expression makes obvious the uselessness of having multiple, successive linear layers, as their combined effect reduces to a single one.
Sentence 2: the dynamics of learning (e.g.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: Then we have a formula for f and a subgradient: f@xA a n k=1 x@kA@f@S k A f @S k 1 AA; @ f@xA Q n k=1 e (k) @f@S k A f @S k 1 AA: Equation (2) was used to show that submodular minimization can be achieved in polynomial time [16].
Sentence 2: algorithms which directly minimize the Lovasz extension are regarded as impractical.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: . . , a n . We will analyze the following popular non-convex objective, (1.1) Previously, Ge et al.
Sentence 2: [GHJY15] show that for the orthogonal case where n ≤ d and all the a i 's are orthogonal, objective function f (•) have only 2n local maxima that are approximately ± 1 the technique heavily uses the orthogonality of the components and is not generalizable to over-complete case.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: 3b shows that the condition number of A becomes worse as the degree d becomes larger, and as more probability mass is assigned to the dense part G d of the transition matrix T , providing some weak evidence for the necessity of Condition 3.
Sentence 2: also, recall that Theorem 1 shows that HMMs where the transition matrix is a random walk on an undirected regular graph with large degree (degree polynomial in n) cannot be learned using polynomially many samples if m is a constant with respect to n. such graphs have all eigenvalues except the first one to be less than O(1/ √ d), hence it is not clear if the hardness of learning depends on the large degree itself or is only due to T being ill-conditioned.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: To date, matrix and tensor decomposition has been extensively analyzed, and there are a number of variations of such decomposition (Kolda and Bader, 2009), where the common goal is to approximate a given tensor by a smaller number of components, or parameters, in an efficient manner.
Sentence 2: despite the recent advances of decomposition techniques, a learning theory that can systematically define decomposition for any order tensors including vectors and matrices is still under development.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The proposed framework is motivated by the operator-theoretic view of nonlinear dynamical systems.
Sentence 2: learning a generative (state-space) model for nonlinear dynamical systems directly has been actively studied in machine learning and optimal control communities, on which we mention a   9) and white Gaussian observation noise and (right) the estimated Koopman eigenvalues.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: First, the accuracy of the belief model β(τ ) in Equation 1 is highly dependent on the performance model P (U |τ, π), which evaluates the policy π behaving against the opponent using policy τ , named response policy.
Sentence 2: the performance model of a response policy against different strategies might be the same in multiagent domains, resulting in indistinguishability of the belief model and thus inaccurate detection.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If G is a tree, this is obviously satisfied.
Sentence 2: the result holds on any graph for which: the subgraph induced by Pu:v is a chain; and every i ∈ Pu:v separates N (i) \ Pu:v from Pu:v \ {i}, where N (i) {j : {i, j} ∈ E} is the neighbor set of i.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: If the algorithm guarantees a zero regret against the competitor with zero L 2 norm, then there exists a sequence of T vectors in X , such that the regret against any other competitor is Ω(T ).
Sentence 2: if the algorithm guarantees a regret at most of > 0 against the competitor with zero L 2 norm, then, for any 0 < η < 1, there exists a T 0 and a sequence of T ≥ T 0 unitary norm vectors z t ∈ X , and a vector u ∈ X such that The proof can be found in the supplementary material.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: The bandit algorithm of Garber (2017) additionally requires a certain boundedness property of barycentric spanners, namely: max q > i Q 2 q i  .
Sentence 2: for certain bounded sets this quantity may be unbounded, such as the two-dimensional axisaligned rectangle with one axis being of size unity, and the other arbitrarily small.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: When computing Precision T (R, P i ) for a single predicted anomaly range P i , there is no need for an existence reward, since precision by definition emphasizes prediction quality, and existence by itself is too low a bar for judging the quality of a prediction (i.e., α = 0).
Sentence 2: the overlap reward is still needed to capture the cardinality, size, and position aspects of a prediction.
Relationship: contrasting
Domain: NeurIPS
Sentence 1: These techniques promote sparsity in determining a small set of codewords from the dictionary that can be used to efficiently represent each visual descriptor of each image [13].
Sentence 2: those approaches consider each visual descriptor in the image as a separate coding problem and do not take into account the fact that descriptor coding is just an intermediate step in creating a bag of codewords representation for the whole image.
Relationship: contrasting