sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We believe a good generative model should be able to synthesize 3D objects that are both highly varied and realistic.
Sentence 2: for 3D objects to have variations, a generative model should be able to go beyond memorizing and recombining parts or pieces from a pre-defined repository to produce novel shapes; and for objects to be realistic, there need to be fine details in the generated examples.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Their main disadvantage is that they do not control the structural complexity of the resulting functions.
Sentence 2: if one of the components of a (say, a weight in a neural network) happens to be irrelevant, a Gaussian prior will not set it exactly to zero, thus The goal of supervised learning is to infer a functional relation ¢¡ ¤£ ¦¥ ¨ § © , based on a set of (maybe noisy) training examples ¡ ¥ ¨ § ¦ © "!
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , F p (X p )), which is specified by the marginal CDFs {F s (X s )} s∈V and the copula density F COP;θ corresponding to the exponential family MRF density.
Sentence 2: the non-parametric extension in (11) of the exponential family MRF densities is precisely an exponential family MRF copula density.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The goal is to establish correspondences between the input-data layers A 0 and A 0 .
Sentence 2: every cell A 0 (x) in the reference object o ∈ Φ has a certain shift d ∈ D in the searched object o ∈ Φ, and we want to estimate d. Here comes the cornerstone idea of our method: we establish the matching of A 0 (x) with A 0 (x − d) for a shift d if there is a pair of "parallel" paths (we call this pair a siamese path), originating at those nodes and ending at the last layers A L , A L , which match.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We focus on learning under a cluster assumption that is formalized in the next section, and establish that there exist nonparametric classes of distributions, denoted P XY , for which the decision sets (over which the target function is smooth) are discernable from unlabeled data.
Sentence 2: moreover, we show that there exist clairvoyant supervised learners that, given perfect knowledge of the decision sets denoted by D, can significantly outperform any generic supervised learner f n in these  Based on this, we establish that there also exist semi-supervised learners, denoted f m,n , that use m unlabeled examples in addition to the n labeled examples in order to estimate the decision sets, which perform as well as f D,n , provided that m grows appropriately relative to n. if the error bound for f D,n decays polynomially (exponentially) in n, then the number of unlabeled data m needs to grow polynomially (exponentially) with the number of labeled data n. We provide general results for a broad range of learning problems using finite sample error bounds.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The population prediction loss γ( Ŷ , n), and group-specific losses γ 0 ( Ŷ , n), γ 1 ( Ŷ , n), for a fixed learning algorithm Ŷ , behave asymptotically as inverse power-law curves with parameters (α, β, δ).
Sentence 2: ∃M, M 0 , M 1 such that for n ≥ M, n a ≥ M a , Intercepts, δ, δ a in (3) represent the asymptotic bias B( ŶD∞ ) and the Bayes error N , with the former vanishing for consistent estimators.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our robustness analysis provides in turn insights onto desirable properties of generative models capturing real-world distributions.
Sentence 2: the intriguing generality of our analysis implies that when the data distribution is modeled through a smooth and generative model with high-dimensional latent space, there exist small-norm perturbations of images that fool humans for any discriminative task defined on the data distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We explore the use of non-parametric models for partially ranked data and derive efficient procedures for their use for large n. The derivations are largely possible through combinatorial and algebraic manipulations based on the lattice of partial rankings.
Sentence 2: we demonstrate for the first time a non-parametric coherent and consistent model capable of efficiently aggregating partially ranked data of different types.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A copy of this MRF is instantiated each time an instance (equivalently, a token, as this relates to the well-known type-token distinction) of X is parsed as v, in the same way that each instance of a symbol in a grammar uses the same categorical distribution to select productions.
Sentence 2: an instance of a symbol has an associated region R ⊆ I and the MRF instantiated for that instance is simply the subset of the full-image MRF containing all of the nodes in R and all of the edges between the nodes in R. The energy of this instance is .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We start by noting that do-calculus [Pearl, 1995] provides rules to evaluate invariances in the interventional space.
Sentence 2: we focus here on the Rule 3, which ascertains the condition such that a set of interventions does not have an effect on the outcome variable, i.e., P (y|do(x, z), w) = P (y|do(x), w).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to remove the equality constraint, which will simplify the problem, let's change variables by defining the n×(n−ν) matrix F as {x : Q T D G x = 0} = {x : x = F y}.
Sentence 2: f is a span for the null space of Q T ; and we will take F to be an orthogonal matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we propose a two-step representation learning method to bridge the feature spaces of different languages by exploiting a set of parallel bilingual documents.
Sentence 2: we first formulate a matrix completion problem to produce a complete parallel document-term matrix for all documents in two languages, and then induce a low dimensional cross-lingual document representation by applying latent semantic indexing on the obtained matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, the bounds for the ensemble Nyström are tighter than those for any Nyström expert based on a single sample of size m even for a uniform weighting.
Sentence 2: for µ = 1/p, the last term of the ensemble bound for norm-2 is smaller by a factor larger than µ max p In this section, we present experimental results that illustrate the performance of the ensemble Nyström method.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the lower bound side, the lower bound result in Blum et al.
Sentence 2: [3] is only for the special case when k = d. We extend their result to every k and d. we show that the sample complexity for collaborative learning has to be Ω(max{d ln k, k ln d}/ ) for constant δ (Theorem 6).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Having observed these performances, the task of predicting the performance of any of them on a new dataset can be cast as a matrix factorization problem.
Sentence 2: we are seeking a low rank decomposition such that Y ≈ XW, where X ∈ R N ×Q and W ∈ R Q×D , where Q is the dimensionality of the latent space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The following local alternative (A1), and balancing assumption (A2), are similar to those used in previous works [6][7][8][9] on problem (1).
Sentence 2: condition (A1) means that the KL-divergence between N (µ 1 , Σ) and N (µ 2 , Σ) tends to 0 as n → ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Proposition 2 The linear machine in ( 7) is equivalent to the following approximate kernel machine Although both random Fourier features and the Nyström method can be viewed as variants of the unified framework, they differ significantly in the construction of the approximate functional space H a .
Sentence 2: the basis functions used by random Fourier features are sampled from a Gaussian distribution that is independent from the training examples.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Because the dimension of V is reduced by one on each iteration of the loop, and because the initial dimension of V is simply the number of selected eigenvectors in step one, the size of Y is distributed as the number of successes in N Bernoulli trials where trial k succeeds with probability λ k λ k +1 .
Sentence 2: |Y | cannot be larger than rank(L), and To get a feel for the sampling algorithm, it is useful to visualize the distributions used to select y i at each time step, and to see how they are influenced by previously chosen items.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Alternatively, we use the importance sampling to estimate the conditional likelihoods [24]: Although the SGVB learning framework has shown to be effective in training deep generative models [16,24], the conditional auto-encoding of output variables at training may not be optimal to make a prediction at testing in deep CGMs.
Sentence 2: the CVAE uses the recognition network q φ (z|x, y) at training, but it uses the prior network p θ (z|x) at testing to draw samples z's and make an output prediction.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To explain, we will consider a simple "toy" data distribution where the behavior of these algorithms is more clear.
Sentence 2: we will generate three heavy-tailed random variables X, Y, Z according to: Here, σ 1 , σ 2 are scale parameters sampled independently from a Laplace distribution, and e 1 , e 2 , e 3 are sampled independently from a unit Gaussian.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition to the distillation loss in KD and the adversarial losses in NaGAN mentioned above, we define a distillation loss from the classifier to the teacher and an adversarial loss between the teacher and the discriminator.
Sentence 2: the classifier and the teacher, serving as generators, aim to fool the discriminator by generating pseudo labels that resemble the true labels.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It was shown in [20]  satisfies the following property: the (pseudo)marginals computed by performing tree-reweighted variational inference with the parameters ✓ := B ⇤ trw ( b ) yield the sufficient statistics b .
Sentence 2: the approximate backward map B ⇤ trw computes an element in the pre-image of the approximate forward map given by tree-reweighted variational inference.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The new solution, denoted by p i+1 , is computed as the projection of p i onto the level set L i , which is equivalent to solving the following optimization problem: Although the projection is regarded as a quadratic programming problem, it can often be solved efficiently because its solution is likely to be the projection onto one of the hyperplanes of polyhedron L i .
Sentence 2: only very few linear constraints of L are active; most of them are inactive.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is sufficient that the objective be polynomial-time evaluable, along with its first and second derivatives, 2 We are obviously assuming X is equipped with an appropriate σ-algebra, and R with the standard Borel σ-algebra, such that the joint distribution P over X × R is well defined and κ(•, •) is measurable.
Sentence 2: 3 let Mn(θ) , and equip the parameter space Θ with the uniform metric • Θ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Glauber dynamics associated with Ising models in high temperature are fast mixing and offer a natural way to define a martingale sequence.
Sentence 2: consider the Doob martingale corresponding to any linear function f for which we wish to show concentration, defined on the state of the dynamics at some time step t  . Also, crucially, t * need not be too large since the dynamics are fast mixing.
Relationship: entailment
Domain: NeurIPS
Sentence 1: These methods are based on using parameterized nonlinear functions to activate each unit in a neural network, with the unit-dependent parameters incorporated into the data-driven training algorithms.
Sentence 2: [6] considered the adaptive piecewise linear (APL) unit defined by a mixture of hinge-shaped functions, and [7] used nonparametric Fourier basis expansion to construct the activation function of each unit.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The FO d-DNNF node ∧ r (n x , n r , D, D , V) represents a decomposable conjunction between the d-DNNF nodes n x , n r and a d-DNNF node isomorphic to the ∧ r node itself.
Sentence 2: the isomorphic operand is identical to the node itself, except for the size of the domain of the variables in V, which becomes one smaller, going from D to D in the isomorphic operand.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A second approach, suggested in [13, p. 34], is to approximate the null distribution as a two-parameter Gamma distribution [12, p. 343, p. 359]: this is one of the more straightforward approximations of an infinite sum of χ 2 variables (see [12,Chapter 18.8] for further ways to approximate such distributions; in particular, we wish to avoid using moments of order greater than two, since these can become expensive to compute).
Sentence 2: we make the approximation An illustration of the cumulative distribution function (CDF) obtained via the Gamma approximation is given in Figure 1, along with an empirical CDF obtained by repeated draws of HSIC b .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then By Corollary 8, for each F r , there exists a partition of V into disjoint sets A r1 , .
Sentence 2: . . , A rMr such that we can write a↵ 0 (B y ) as the nullspace of either of the matrices . . . . . . , where 1 A is the indicator vector of A ✓ V . For T 0 , this follows directly from Equation ( 6).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We call an optimization method rescaling invariant if the updates of rescaling equivalent networks are rescaling equivalent.
Sentence 2: if we start at either one of the two rescaling equivalent weight vectors w(0) ∼ w (0) , after applying t update steps separately on w(0) and w (0) , they will remain rescaling equivalent and we have w(t) ∼ w (t) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The ex ante coordinated strategy profile for the team is implicitly represented by the pair ( λ, ωT ,σ ).
Sentence 2: that pair encodes a coordination device that operates as follows: • At the beginning of the game, a pure normal-form plan σ ∈ Σ is sampled according to the discrete probability distribution encoded by λ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Furthermore, the performance prediction loss acts as a regularizer that forces the encoder not optimized into a trivial state to simply copy tokens in the decoder side, which is typically eschewed by adding noise in encoding x by previous works [1,24].
Sentence 2: when both the encoder and decoder are optimized to convergence, the inference process for better architectures is performed in the continuous space E. starting from an architecture x with satisfactory performance, we obtain a better continuous representation e x by moving e x = {h 1 , • • • , h T }, i.e., the embedding of x, along the gradient direction induced by f : where η is the step size.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A popular family of learning algorithms called temporal difference (TD) methods [1] are designed for this situation.
Sentence 2: least-squares TD (LSTD) algorithms [2,3,4] exploit the linearity of the value function to estimate its parameters from sampled trajectories, i.e., from sequences of feature vectors of visited states, by solving a set of linear equations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Figure 2 reports numerical results obtained.
Sentence 2: leftmost plots report results associated with models trained on sequences from the Highway split and tested on the New York-like City and the Old European Town splits (top-left and bottom-left, respectively); rightmost plots report results associated with models trained on sequences from the New York-like City split and tested on the Highway and the Old European Town splits (top-right and bottom-right, respectively).
Relationship: entailment
Domain: NeurIPS
Sentence 1: At the beginning of every epoch k, for every s ∈ S, a ∈ A such that N s,a ≥ η, it generates multiple samples for P s,a from a 'boosted' posterior.
Sentence 2: it generates ψ = O(S log(SA/ρ)) independent sample probability vectors , where M t s,a denotes the vector [M t s,a (i)] i=1,...,S , with M t s,a (i) := 1 κ (N t s,a (i) + ω), for i = 1, . . . , S. (3) Here, κ = O(log(T /ρ)), ω = O(log(T /ρ)), η = T S A + 12ωS 2 , and ρ ∈ (0, 1) is a parameter of the algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As shown in Table 2, for the random distractor, the 'EM Model' substantially outperforms the 'NaIve Model' in all cases.
Sentence 2: for the 'wine' and 'iris' datasets, by introducing an additional class label to every training instance, there is only one class label left out of the class label candidates and yet the performance of the 'EM Model' is still close to the case when there are no additional class labels.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus we propose to automatically fill the missing entries of M 0 based on the feature co-occurrence information expressed in the observed data, by conducting matrix completion to recover a low-rank and sparse matrix.
Sentence 2: we formulate the matrix completion as the following optimization problem where • 1 denotes a ℓ 1 norm and is used to enforce sparsity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, for any γ > 0, the sample complexity of dense grid features can be bounded by .
Sentence 2: as was the case with polynomially-exact features, for a fixed d, D( Unfortunately, this scheme suffers heavily from the curse of dimensionality, since the sample complexity is doubly-exponential in d. This means that, even though they are easy to compute, the dense grid method does not represent a useful solution to the issue posed in Section 3.2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This set contains 675,000 galaxies observed in D = 3750 spectral bins, preprocessed by first moving them to a common rest-frame wavelength and filling-in missing data following [18] but using the more sophisticated weighted PCA algorithm of [5], before computing a sparse neighborhood graph and pairwise distances between neighbors in this graph.
Sentence 2: a log-log plot of the average number neighbors m(r) vs. neighborhood radius r (shown in the Supplement), indicates that the intrinsic dimension of these data varies with the scale r. in order to support m = O(d) neighbors, the radius must be above 60, in which case d ≤ 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that other values of κ > 1 can be used, but the choice κ = 2 given above results in simpler semantics.
Sentence 2: if a compensation happens to satisfy c-map(X i = x) = c-map(X j = x) = c-map(X i = x, X j = x) for some state x, we have that θ(X i = x) + θ(X j = x) = map(X i = x, X j = x) (i.e., the parameters alone can recover an original MAP value).
Relationship: entailment
Domain: NeurIPS
Sentence 1: * y) + where ∼ N (0, σ 2 1) , z ∼ IBP(α) and y i ∼ p(y) Again, the goal is to infer A, the probabilities π i and then to associate likely values of Z ij and Y ij with the data.
Sentence 2: [24] make a number of alternative assumptions on p(y), namely either that it is iid Gaussian or that it is iid Laplacian.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our case of a binary sample space, this equates to selecting one random axis h ∈ {1, . . . , H} and toggling its bit value (thereby changing the binary state in that dimension), leaving the remaining axes unchanged.
Sentence 2: the posterior probability computed for each candidate sample is expressed by: where we have introduced a parameter β that allows for smoothing of the posterior distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, as there are r = 20✏ 1 log 1+✏/4 (n) iterations of BLITS, the adaptivity is O log 2 n . Our goal in this section is to show that beyond its provable guarantees, BLITS performs well in practice across a variety of application domains.
Sentence 2: we are interested in showing that despite the fact that the parallel running time of our algorithm is smaller by several orders of magnitude than that of any known algorithm for maximizing non-monotone submodular functions under a cardinality constraint, the quality of its solutions are consistently competitive with or superior to those of state-of-the-art algorithms for this problem.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This can be done by assigning a prior of zero to any ✓ yielding ln Now, using Equation (7) in the optimal posterior (Equation 4) simplifies to where the normalization constant Z X,Y corresponds to the Bayesian marginal likelihood: This shows that the optimal PAC-Bayes posterior given by the generalization bound of Theorem 1 coincides with the Bayesian posterior, when one chooses `nll as loss function and := b a (as in Equation 2).
Sentence 2: moreover, using the posterior of Equation ( 8) inside Equation ( 3), we obtain = n minimizing the PAC-Bayes bound is equivalent to maximizing the marginal likelihood.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our analysis shows that EMGD is able to find an ǫ-optimal solution by computing O(log 1 ǫ ) full gradients and O(κ 2 log 1 ǫ ) stochastic gradients.
Sentence 2: with the help of stochastic gradients, the number of full gradients required is reduced from O( √ κ log 1 ǫ ) to O(log 1 ǫ ), independent from the condition number.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let ε be a given error parameter.
Sentence 2: now, using the definition of the g (t) functions, and by the regret bound for the EG algorithm, we have that for any fixed distribution α ∈ ∆ N , Since ∇f (t) ≤ 1, Summing (4) from t = 1 to T , and adding to (3), we get that for any distribution α over Φ, by concentrating α on any given φ i , the above inequality implies that T t=1 f (t) (φ i (x (t) )) − T t=1 f (t) (x (t) ) ≤ O( log(N )T ), and thus we have a no Φ-regret algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume X is specified in a way so that the algorithm can efficiently construct the smallest Euclidian ball containing X .
Sentence 2: furthermore, we assume the algorithm has noisy black-box access to f . the algorithm is allowed to query the value of f at any x ∈ X , and it observes y = f (x)+ε where ε is an independent σ-subgaussian random variable with mean zero: E[exp(λε)] ≤ exp(λ 2 σ 2 /2) for all λ ∈ R. The goal of the algorithm is to minimize its regret: after making T queries x 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Due to the wide applicability of this class of M -estimators in (1), there has been a line of work on developing efficient optimization methods for solving special instances of this class of M -estimators [14,26], in addition to the papers listed above.
Sentence 2: due to the superposition-structure in (1) and the high-dimensionality of the problem, this class seems naturally amenable to a proximal gradient descent approach or the ADMM method [2,17]; note that these are first-order methods and are thus very scalable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our main interest is in the local discrepancy function disr S : R d → R parametrised by a given set of samples S = {X 1 , ..., X |S| } and defined as follows: where:  where λ ∈ P(R 1 ), and a set of samples S = {X 1 , ..., X |S| } that is used in a given (Q)MC estimator to approximate I f . The star discrepancy function D * λ with respect to a distribution λ is defined on S as: ..,|S| and F λ stands for the cdf function for λ.
Sentence 2: to measure the discrepancy between arbitrary distribution λ ∈ P(R 1 ) and a set of samples S, the set of samples is transformed to the interval [0, 1] via the cdf F λ and the discrepancy between the uniform distribution on [0, 1] and the transformed sequence F λ (S) is calculated.
Relationship: entailment