sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Most of these applications reduce to some instance of a tensor decomposition problem, and the growing interest in joint matrix decomposition is largely motivated by such reductions.
Sentence 2: in the past decade several 'matricization' methods have been proposed for factorizing tensors by computing the joint decomposition of sets of matrices extracted from slices of the tensor (see, e.g., [10,22,17,8]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: If p ≥ n then the set of solutions is convex, with the value of ∇ θ f ( θ; x) constant across all solutions.
Sentence 2: if p ≥ n and |∇ θt f ( θ; x)| < λ for some solution θ, then θ t = 0 for all solutions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consequently, we can write f = 1 T d + K 1 e, and so our formulation becomes: The solution to this optimization problem is for our coefficients d and e to be linear estimates on y, our vector of observed values.
Sentence 2: there exist matrices Υ d (c, m) and Υ e (c, m), dependent on our smoothing coefficient c and our exponent m, such that: c, m) is the influence matrix [9] which provides the function estimate over the entire graph.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For sub-exponential design and noise, the difference compared to the sub-Gaussian case is the dependence on the exponential width instead of the Gaussian width of the unit norm ball.
Sentence 2: using known results on the Gaussian widths of unit 1 and group-sparse norms, corollaries below are derived using the relationship between Gaussian and exponential widths derived in Section 3: Corollary 2 If R(•) is the group-sparse norm, for sub-exponential design matrix X and noise ω, For Gaussian and sub-Gaussian X, previous work has established RIP bounds of the form RIP is satisfied w.h.p if the number of samples is of the order of square of the Gaussian width of the error set ,i.e., O(w 2 g (A)), which we will call the sub-Gaussian RE sample complexity bound.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The high-level view of the above result is that choosing support points y i to form an ✏-cover with respect to the metric g, and then optimizing over their weights w i via our stochastic algorithm, will give us a consistent picture of the behavior of the true barycenter.
Sentence 2: also note that the proof above requires an ✏-cover only of the support of v ⇤ , not all of K. an ✏-cover of the convex hull of the supports of µ j is sufficient, as this must contain the barycenter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The priors select some large scale features that are the most important to learn first and fill in the details as more data become available (see [11]  One can summarize this in an Occam-like fashion [11]: if two models provide equally good fits to data, a simpler one should always be used.
Sentence 2: the predictive information, which quantifies complexity [11], and of which A is the derivative, in a QFT model is ......, N 1 / 2 TJ, and it is . . . . . . , kc log N in the parametric case.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, while most approximation algorithms output an approximation of the optimum value of the linear program (1), we also describe a simple, parallelizable rounding algorithm that provably outputs a feasible solution to (1).
Sentence 2: for any ε > 0 and bounded, non-negative cost matrix C, we describe an algorithm that runs in time O(n 2 /ε 3 ) and outputs P ∈ U r,c such that P , C ≤ min P ∈Ur,c We emphasize that our analysis does not require the cost matrix C to come from an underlying metric; we only require C to be non-negative.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since set F does not exist in input networks, one can sample α target nodes k (with probability proportional to in-degree m k ) to form negative examples (i, k) .
Sentence 2: given source node i, we emphasize the existence of link (i, j) by distinguishing whether the corresponding target node is observed ((i, j) ∈ E) or not ((i, k) ∈ F ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: If we assume that the body may be modeled as a kinematic tree, and motion of a particular rigid segment is known, then the motions of the rigid segments that are connected through that segment are independent of each other.
Sentence 2: we can model a probability distribution of the full body-pose as a tree-structured graphical model, where each node corresponds to pose of a rigid segment.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Most related work is based on the strong and limiting assumption that Z is a binary random variable (e.g., Z = 0 for the source domain, and Z = 1 for the target domain).
Sentence 2: (Pan et al., 2011;Gong et al., 2013;Baktashmotlagh et al., 2013;Zemel et al., 2013;Ganin and Lempitsky, 2014;Ajakan et al., 2014;Edwards and Storkey, 2015;Louizos et al., 2015) are all based on the minimization of some form of divergence between the two distributions of f (X)|Z = 0 and f (X)|Z = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given the data available for both tasks, our objective is to build a good estimate for the conditional density p t (y|x).
Sentence 2: to address this domain adaptation problem, we assume that p t is a modified version of p s . we assume that p t is obtained in two steps from p s . First, p s is expressed using an R-vine representation as in (10) and second, some of the factors included in that representation (marginal distributions or pairwise copulas) are modified to derive p t . All we need to address the adaptation across domains is to reconstruct the R-vine representation of p s using data from the source task, and then identify which of the factors have been modified to produce p t . These factors are corrected using data from the target task.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Quite interestingly, because of the algebraic structure we designed for the computation strategy (i.e., equation ( 10)), the decoding process can be viewed as a polynomial interpolation problem (or equivalently, decoding a Reed-Solomon code).
Sentence 2: in this example each worker i returns which is essentially the value of the following polynomial at point x = i: Hence, recovering C using computation results from 4 workers is equivalent to interpolating a 3rddegree polynomial given its values at 4 points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The sequence (ϑ t ) t≥1 in Theorem 3 is not a parameter of the algorithm, but rather a parameter in the regret bound.
Sentence 2: (11) holds true for any such sequence, and we will use this fact later on to obtain explicit bounds by instantiating (11) with a particular choice of (ϑ t ) t≥1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: An important distinction between our model and Fisher's, including its graphical extension [16], is that our model allows buyers (i.e., villages) to invest the goods (i.e., loans) in productive projects, thereby generating revenue that can be used to pay for the goods (i.e., repay the loans).
Sentence 2: the crucial modeling parameter of "endowment" is no longer a constant in our case.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our work distinguishes itself from these directions mainly in the objective, since we require the non-discrimination to happen in the long-term instead of at any time; this extends the classical batch definitions of non-discrimination in the online setting.
Sentence 2: we only focus on situations where there are enough samples from each population of interest and we do not penalize the algorithm for a few wrong decisions, leading it to be overly pessimistic.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The domain is the entire Banach space W = p . Then, for any δ > 0 and any a > 0, with probability at least 1 − δ over a sample of size n, we have that for all w ∈ W = p (of any magnitude): Corollary 2 allows us to analyze the rate of convergence of the regularized risk for p -regularized linear learning.
Sentence 2: training by minimizing the empirical average of: where (z, y) is some convex loss function and x q ≤ B.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a network using simulation-on-the-fly, 20000 training batches were run and evaluated on the same test set.
Sentence 2: we ran both the simulation on-the-fly and fixed training set for the same number of iterations with a batch size of 50, but the simulation-on-the-fly draws a fresh datapoint from the generative model upon each update so that no datapoint is used more than once.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to perform cost-sensitive exploration in the Bayesian RL (BRL) setting, we cast the problem as a constrained partially observable MDP (CPOMDP) [11,12] planning problem.
Sentence 2: we take a model-based BRL approach and extend BEETLE [4] to solve the CPOMDP which models BRL with cost constraints.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (Lemma 1 from [6]) Let k : X × X → R be a universal kernel on a compact subset X or R d and Φ : X → H be a feature map of k. The Φ is continuous and defines a metric on X such that id : (X, • ) → (X, d k ) is continuous.
Sentence 2: n ((X, d k ), ε) is finite for all ε > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider two possible profiles of reward distributions, ν and ν , that are "close" enough in the sense that it is hard to distinguish between the two, but "separated" enough such that a considerable regret may be incurred when the "correct" profile of distributions is misidentified.
Sentence 2: assume that the decision maker is a priori informed that the first arm generates rewards according to a normal distribution with standard variation σ and a mean that is either −∆ (according to ν) or +∆ (according to ν ), and the second arm is known to generate rewards with normal distribution of standard variation σ and mean zero for both ν, and ν .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our next step involves generalizing the earlier proposition to multivariate symmetric functions by representing them as tensors, which then yields a corresponding spectral theorem of product decompositions for such functions.
Sentence 2: note that when the feature vector of each document takes values only from a finite set X , of size |X |, a symmetric function h(x 1 , x 2 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Interestingly, if γ > 1 and 0 < T < 1, then LoSST is neither convergent nor diligent, but rather strikes a balance between the desiderata.
Sentence 2: these versions (a) tend to converge towards stable structures, but provably do not actually converge since they remain sensitive to outliers; and (b) respond quickly to change in generating structure, but only exponentially fast in the number of previous datapoints, rather than within a fixed interval.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To include the information of observable pattern methods in the factorization, we augment the RESCAL model with an additive term that holds the predictions of observable pattern methods.
Sentence 2: let X P t0, 1u N ˆN ˆK be a third-order adjacency tensor and M P R N ˆN ˆP be a third-order tensor that holds the predictions of an arbitrary number of relational learning methods.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our more formal model provides interesting insights into learned Bloom filters, and how they might be effective.
Sentence 2: here we: (1) clarify what guarantees can and cannot be associated with such a structure; (2) show how to estimate what size the learning function must obtain in order to obtain improved performance; (3) provide a simple method for optimizing learned Bloom filters; and (4) demonstrate our approach may be useful for other similar structures.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, we propose the following class of estimators: where g λ (•) is a filter function and λ is referred to as a shrinkage parameter.
Sentence 2: the matrix-valued function g λ (K) can be described by a scalar function g λ : [0, κ 2 ] → R on the spectrum of K. if K = UDU ⊤ is the eigen-decomposition of K where D = diag(γ 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , (x m , y m ), the average training loss of a matrix W is: L(W ) = 1 m P (x,y)2S `(W, (x, y)).
Sentence 2: we aim at approximately solving the problem find the matrix W with minimal training loss among all matrices with column sparsity of at most s, where s is a user-defined parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let M be a model with hidden and visible variables z = (x, y) that satisfy a Markov network factorisation.
Sentence 2: the joint density can be written as a product of clique-potentials 'lj;j, P(zI9) = g(9) TI j 'lj;j (Cj ,9), where each clique Cj is a subset of the variables in z.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As the main result, we provide finite sample analysis of NNQL for a single, arbitrary sequence of data for any infinite-horizon discounted-reward MDPs with continuous state space.
Sentence 2: we show that the algorithm outputs an "-accurate (with respect to supremum norm) estimate of the optimal Q-function with high probability using a number of observations that depends polynomially on ", the model parameters and the "cover time" of the sequence of the data or trajectory of the data utilized.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given the apparent difficulty of learning models that are robust to adversarial perturbations, we propose tackling the simpler problem of developing adversarially robust features.
Sentence 2: given a dataset and metric of interest, the goal is to return a function (or multiple functions) that 1) is robust to adversarial perturbations, and 2) has significant variation across the datapoints.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 1 and KL Compute output dist Backprop ∇ θ and ∇ φ , reparam and KL inference methods directly aim to tighten this gap.
Sentence 2: the evidence lower bound (ELBO) is a parameterized bound over a family of distributions q(z) ∈ Q (with the constraint that the supp q(z) This allows us to search over variational distributions q to improve the bound.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is possible to show that the isd posterior is marginally consistent at least in the Gaussian mean case (one element of the exponential family).
Sentence 2: marginalizing over an observation and its associated marginal's parameter (which can be taken to be x N and θ N without loss of generality) still produces a similar isd posterior on the remaining observations X /N and parameters Θ /N .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Reweighting-based strategies use inverse-probability weighting (IPW) to change measure from the outcome distribution induced by a logging policy to that induced by the policy ⇡.
Sentence 2: these methods use the fact that, under unconfoundedness, V IPW (⇡) is unbiased for V (⇡) [15], where Optimizing V IPW (⇡) can be phrased as a weighted classification problem [3].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we focus on the large-scale graph scenario when the dimension d is large and hence assume the true precision matrix to be sparse such that Θ 1 ≤ L. We formulate a procedure for testing Θ ∈ S using an estimated precision matrix Θ obtained from samples X 1 , .
Sentence 2: . . , X n . Θ can be estimated via the node-wise regression [21], the CLIME [5], or the graphical lasso [11].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The subtlety in proving these results is that there could exist false mixtures that agree with the observed entries, even if the sampling is uniquely completable for LRMC and HRMC (see Example 1).
Sentence 2: there exits samplings that are identifiable for LRMC (and HRMC) but are not identifiable for MMC, and so in general it is not enough to simply have K times more samples.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we develop a convex formulation of multi-layer learning that allows multiple latent kernels to be connected through nonlinear conditional losses.
Sentence 2: each pair of successive layers is connected by a prediction loss that is jointly convex in the adjacent kernels, while expressing a non-trivial, non-linear mapping between layers that supports multi-factor latent representations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The most useful versions of Wasserstein distance are p = 1, 2, with p = 1 being the weaker of the two (by Hölder's inequality, p ≤ q ⇒ W p ≤ W q .)
Sentence 2: "results in W 2 distance are usually stronger, and more difficult to establish than results in W 1 distance" [35, p. 95].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Real data (Figure 1) provides evidence that users tend to be approximately consistent and therefore simpler patterns, i.e., patterns that mostly stick to a single rounded value y(t) = y correspond to larger sets of sequences {x i (t)} t , obtained from a real population.
Sentence 2: for each app there is always one pattern (corresponding to having one fixed y(t) = y across all 31 days) which blends the majority of users (> 2 million).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Firstly, we iteratively apply the model to predict one more time step into the future by treating the prediction as input in a recursive way.
Sentence 2: for predicting multiple time steps in the flow anticipating network, we warp the most recent frame X t−1 using the output prediction Ôt to get the Xt which is then combined with X t−k−1:t−1 to feed the flow anticipating network to generate Ôt+1 , and so forth.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The purpose of this paper is to investigate how ranking methods for multivariate data with binary labels may be exploited in order to extend the rank-based test approach for testing homogeneity between two samples to a multidimensional setting.
Sentence 2: the testing principle promoted in this paper is described through an extension of the Mann-Whitney Wilcoxon test, based on a preliminary ranking of the data through empirical AUC maximization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Joint object/feature sparsity via 1 / 2 + 1 (sparse group) regularized LR (LRG1) The groupregularized LR does not, however, yield sparsity within a group (object) for those groups with nonzero total weights.
Sentence 2: if a group of parameters is non-zero, they will all be non-zero.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More generally, if y is discrete and T (x, y) consists of |Y| repetitions of a fixed feature function t(x) (as in Figure 1a), then we can write Provided η ⊤ y t(x) is convex in x for each η y , the level sets of A as a function of x form the boundaries of convex sets.
Sentence 2: exactly normalizable sets are always the boundaries of convex regions, as in the simple example Figure 1a.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2, left, bottom) reveals that smoothing actually helps to obtain a slightly better solution here.
Sentence 2: the soft-max CD (with = 0.001) and L 2 -max AGD (with = 0.01), as well as the primal (SDCA) and dual (AGD) algorithms for the smooth and strongly convex objective, are able to recover the optimal solution within the allowed number of iterations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead, as illustrated in Figure 1 (left), they rely on hidden states H l−1 t and cell states M l−1 t (l ∈ {1, ..., L}) that are updated by the previous layer at current time step.
Sentence 2: the bottom LSTM unit receives state values from the top layer at the previous time step: The four layers in this figure have different sets of input-to-state and state-to-state convolutional parameters, while they maintain a spatiotemporal memory cell and update its states separately and repeatedly as the information flows through the current node.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The reactions used for training are atom-mapped so that each atom in the product graph has a unique corresponding atom in the reactants.
Sentence 2: reaction Center A reaction center is a set of atom pairs {(a i , a j )}, where the bond type between a i and a j differs from G r to G p . a reaction center is a minimal set of graph edits needed to transform reactants to products.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We solve the two-sample testing problem when the inputs are two CBNs over the same DAG G in n variables; for a given > 0 and DAG G, call the problem C2ST(G, ).
Sentence 2: we show an algorithm to solve C2ST(G, ) that makes O(log n) interventions on the input models X and Y, uses O(n/ 2 ) samples per intervention and runs in time Õ(n 2 / 2 ), when G has bounded in-degree and c-component size.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, for each Ü we define a conditional distribution over points such that If all points in the same class were mapped to a single point and infinitely far from points in different classes, we would have the ideal "bi-level" distribution: Furthermore, under very mild conditions, any set of points which achieves the above distribution must have the desired geometry.
Sentence 2: assume there are at least Ö • ¾ points in each class, where Ö rank ℄ (note that Ö Ö).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work we utilize the channel features detectors [9, 7, 1, 2], a family of conceptually straightforward and efficient detectors for which variants have been utilized for diverse tasks such as pedestrian detection [10], sign recognition [22] and edge detection [19].
Sentence 2: for our experiments we focus on pedestrian detection and employ the aggregate channel features (ACF) variant [7] for which code is available online 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Intuitively, a model with default values is used to encode the first few data; then the model is trained on these few encoded data; this partially trained model is used to encode the next data; then the model is retrained on all data encoded so far; and so on.
Sentence 2: we call p a prediction strategy for predicting the labels in Y knowing the inputs in X if for all k, p(y k+1 |x 1:k+1 , y 1:k ) is a conditional model; namely, any strategy for predicting the k + 1label after already having seen k input-output pairs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Moreover, assume that (RI), (3.2) and (3.3) hold.
Sentence 2: then for all k large enough, If ρ(M ) < 1, then given any ρ ∈]ρ(M ), 1[, there exists K ∈ N such that ∀k ≥ K, if s = 1, then ρ(M ) < 1 if R is locally polyhedral around x or if a 0 = b 0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper develops (conic) geometric optimisation on the cone of hpd matrices, which allows us to globally optimise a large class of nonconvex functions of hpd matrices.
Sentence 2: we first use the Riemannian manifold structure of the hpd cone for studying functions that are nonconvex in the Euclidean sense but are geodesically convex (g-convex), hence globally optimisable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we aim at finding an interpretable perturbation δ ∈ X ∩ x 0 such that after removing it from x 0 , arg max i [Pred(x 0 )] i = arg max i [Pred(δ)] i .
Sentence 2: x 0 and δ will have the same top-1 prediction class t 0 , indicating that the removed perturbation δ is representative of the model prediction on x 0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Suppose that on any bounded subset of the domain of F , it holds that F θ (w, V ), ∇F θ (w, V ) and ∇ 2 F θ (w, V ) are all Lipschitz continuous in (w, V ).
Sentence 2: then every local minimum (w, V, θ) of F satisfies the objective F has no spurious local minima with value above the smallest attainable with a linear predictor.
Relationship: entailment