sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: This image is distorted by variations between the properties of individual target cell and neurotropin circuits that are introduced during fabrication.
Sentence 2: poor power distribution on the Neurotrope1 chip creates a systematic gradient in target cell excitability that warps a growth cone's impression of the relative coactivity of its neighbors, attracting it preferentially toward the more excitable target cells on the right side of the array.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, their method requires multiple passes over the data, thus doesn't fit directly in the streaming k-means setting.
Sentence 2: the above mentioned paper analyzes the stochastic k-means method only for highly accurate initial set of iterates which requires a large burn-in period of t = O(N 2 ) and hence needs O(N ) passes over the data, where N is the number of data points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: His revenue resulting from the cleared bid will be the good's spot price (utility) π t,k .
Sentence 2: the payoff obtained from good k at period t is (π t,k − λ t,k )1{x t,k ≥ λ t,k } where 1{x t,k ≥ λ t,k } indicates whether the bid is cleared.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Counterfactual fairness also provides an answer to some problems on the incompatibility of fairness criteria.
Sentence 2: consider the following problem raised independently by different authors (e.g., [7,22]), illustrated below for the binary case: ideally, we would like our predictors to obey both Equality of Opportunity and the predictive parity criterion defined by satisfying as well as the corresponding equation for Ŷ = 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using F to denote the first node ordered in the contamination set of (s → t) (and L for the last), we can compute C[1, N ] via the following recursive update principle where the maximization is with respect to the structure score (e.g., BIC).
Sentence 2: the best chain in a subsequence [i, j] in the ordering is the maximum of three alternatives: edges whose contamination boundaries are exactly i and j (no split); two chains that are joined at some node i < k < j (split); a gap between i and j when there is no positive edge whose contamination is in Finally, for lack of space we only provide a brief description of our topological node ordering.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For any p > 0, there exist constants (C 0 > 0, ρ s > 0, m 0 ) with the following property: if m > m 0 , (A 0 , E 0 ) ∈ R m×m × R m×m with the singular spaces of A 0 ∈ R m×m distributed according to the random orthogonal model of rank and the signs and support of E 0 ∈ R m×m distributed according to the Bernoulli sign-and-support model with error probability ≤ ρ s , then with probability at least 1 − Cm −p and the minimizer is uniquely defined.
Sentence 2: matrices A 0 whose singular spaces are distributed according to the random orthogonal model can, with probability approaching one, be efficiently recovered from almost all corruption sign and support patterns without prior knowledge of the pattern of corruption.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [16] in the context of kernel Fisher's discriminant analysis contains the key idea that we will follow: noise and distracting signal aspects with respect to which we want to make our feature extractor invariant is added to the denominator of a Rayleigh coefficient.
Sentence 2: our prior knowledge about the noise type helps to re-design the optimization of CSP feature extraction.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The stronger distributional assumptions made in the generative approach result in a less robust learner, but make it possible to learn linguistic exceptions without negative evidence.
Sentence 2: generative models can exploit the "implicit negative evidence" provided by the absence of a sentence: the assumption that sentences are generated from the target probability distribution means that not observing a sentence provides weak evidence that it does not belong to the language.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We are required to produce a prediction, ŷt ∈ [k], as the label of x t . In response, we obtain only 1 bit of information: whether ŷt = y t or not.
Sentence 2: when ŷt = y t , the identity of y t remains unknown (although one label, ŷt , is ruled out).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To produce Figures 1a and 1b, we reimplemented the toy experiment of Bishop [5,Section 3.5.1].
Sentence 2: we generated a learning sample of 15 data points according to y = sin(x) + ✏, where x is uniformly sampled in the interval [0, 2⇡] and ✏ ⇠ N (0, 1 4 ) is a Gaussian noise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To aid the presentation below we simplify this model a bit further.
Sentence 2: set b = 0 and observe that (7) reduces to where γ k = −r − 2c ψ k . we modify the parameterization to θ = {γ k } k=1 ∪ {θ P AQ } such that θ P AQ denotes the matrix of parameters in (8).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We then apply this result to obtain several new results related to subset ranking.
Sentence 2: we first obtain calibrated, score-based surrogates for the Precision@q loss, which includes the winner-take-all (WTA) loss as a special case, and the expected rank utility (ERU) loss; to the best of our knowledge, consistency with respect to these losses has not been studied previously in the literature.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is seen that the margin l 1 and l 2 models perform better than cross-entropy across the entire range of noise levels, while the margin l ∞ model is slightly worse than cross-entropy.
Sentence 2: the margin l 2 model achieves a evaluation accuracy of 96.4% at 80% label noise, compared to 93.9% for cross-entropy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Rather than trying to learn a robust function that also achieves high classification accuracy, we consider the problem of learning any function that is robust to adversarial perturbations with respect to any specified metric.
Sentence 2: given a dataset of d-dimensional points and a metric of interest, can we learn features-namely functions from R d → R-which 1) are robust to adversarial perturbations of a bounded magnitude with respect to the specified metric, and 2) have significant variation across the datapoints (which precludes the trivially robust constant function).
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, when p(y t |x t ) is twice differentiable, it is straightforward to find an approximating Gaussian model based on a Laplace approximation by simple numerical optimization [12,33,30], and use the obtained model as a basis of twisted SMC.
Sentence 2: we use where p(y t |x t ) ≈ p(y t |x t ), t = 1, . . . , T are the Gaussian approximations obtained using Laplace's method.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We propose to use a primal-dual algorithm for the inner problem and show experimentally that this approach yields faster convergence than the FISTA method of [5] which was applied in [14].
Sentence 2: let us restrict our attention to the case where R(f we apply the RatioDCA algorithm to (2) with S = S 1 which is what we need, e.g., for the tight relaxations of the Cheeger cut, normalized cut and truncated Cheeger cut families.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (iii) The columns of E 3 fall in the column span of C and tr(E (iv) The rows of E 4 fall in the row span of C and tr(E and 1 + 2 + ˜ 2 + 3 + 4 = , then X is a -approximation embedded matrix for X.
Sentence 2: we have The proof can be referred to [17].
Relationship: entailment
Domain: NeurIPS
Sentence 1: G is defined as For any hypothesis class H, target hypothesis h ∈ H, distribution P , sample S m , and real r > 0, define Finally, for any h ∈ H we define a ball in H of radius r around h [5].
Sentence 2: with respect to class H, marginal distribution P over X , h ∈ H, and real r > 0, define The concept of perfect classification was introduced in [1] within a realizable selective classification setting.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 3.1 is stated as a bound on the probability that the norm of the error matrix is greater than some target error " given the number of samples s. In practice, the target error " is typically not known in advance, but rather is the quantity to minimize, given the matrix A, the number of samples s, and the target confidence .
Sentence 2: for any given distribution p on the elements of A, define Our goal in the rest of the paper is to seek the distribution p minimizing " 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In general, a graph-based method wants to find a function to simultaneously satisfy two conditions [2]: a) it should be close to given labels on the labeled nodes, and b) it should be smooth on the whole graph.
Sentence 2: the work in [8] develops a regularization framework to carry out the above idea by defining the global and local consistency terms in their cost function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When the Gibbs sampler chooses to resample a cluster C that covers D (the input of f ), we can update the Rao-Blackwellized estimator for free.
Sentence 2: the Gibbs sampler computes the cluster belief ) in order to compute the transition density p C∩E|E\C (• | w n−1 E\C ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to incorporate the label information, we add a semi-supervised training phase.
Sentence 2: we set the number of clusters to be the same as the number of class labels and after executing the reconstruction and the adversarial phases on an unlabeled mini-batch, the semi-supervised phase is executed on a labeled mini-batch, by updating the weights of the encoder q(z|x) to minimize the cross-entropy cost.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While FastRNN controls the condition number of gradient reasonably well, its expressive power might be limited for some datasets.
Sentence 2: this concern is addressed by a novel architecture, FastGRNN, that uses a scalar weighted residual connection for each and every coordinate of the hidden state h t . where 0 ≤ ζ, ν ≤ 1 are trainable parameters that are parameterized by the sigmoid function, and σ : R → R is a non-linear function such as tanh, sigmoid and can vary across datasets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Basically, each layer is processed independently based on the initial model input and output, without taking into account the retraining result from the previous layer.
Sentence 2: denoting Y ( −1) and Y ( ) as the input and output of the -th layer of the initial trained neural network (see equation ( 1)), we propose to relearn the coefficient matrix W via the convex program The optimization in (5) can be independently applied to every layer in the network and hence computationally distributable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We define a weighted and directed graph G UC on all clusterings C. Thus, N = B n ≤ n n is the n th Bell number.
Sentence 2: when C is obtained by a MERGE of two clusters in C, G UC contains a directed edge G UC contains an edge from C to the clustering obtained from breaking C i into singleton clusters of all its elements.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In other words, ⇡(i, s, k) corresponds to a choice probability when restricting U to the first k products, a 1 , .
Sentence 2: . . , a k . With this notation, we have for all i 2 [n], P(a i |S) We compute ⇡(i, s, k) iteratively for k = 1, . . . , n. in order to compute ⇡(i, s, k + 1), we use the correctness of the sampling procedure.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since our algorithm does not depend on ℓ, our analysis is actually true for any semimetric ℓ that satisfies Assumptions 1-4, thus Theorem 2 and Corollary 2 hold for the best possible choice of such a ℓ.
Sentence 2: we can think of problems for which there exists a semimetric ℓ such that the corresponding near-optimality dimension d is 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our goal is to compute the top k principal components of the data: the k-dimensional subspace that offers the best squared-error estimate for the points.
Sentence 2: we assume a probabilistic generative model, from which the data is sampled at each step t. where A ∈ R p×k is a fixed matrix, z t ∈ R k×1 is a multivariate normal random variable, i.e., and w t ∈ R p×1 is the "noise" vector, also sampled from a multivariate normal distribution, i.e., Furthermore, we assume that all 2n random vectors (z t , w t , ∀1 ≤ t ≤ n) are mutually independent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We assume that our bandit algorithm receives a context x t ∈ X at each round t, and that there exists a function f ∈ F, in some known concept class F, such that f (x t ) = +1 if an event occurs at round t, and f (x t ) = −1 otherwise.
Sentence 2: 2 f is an event oracle.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Recent papers further show that this problem can be formulated and solved as a compressed sensing problem using random queries [14,18].
Sentence 2: [14] gives an algorithm using O(s 2 n) queries based on mutual coherence, whereas the Restricted Isometry Property (RIP) is used in [18] to give a query complexity of O(sn 4 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: A column vector of ones of arbitrary dimension will be denoted by e. For A ∈ ℜ m×n and B ∈ ℜ n×k , the kernel K(A, B) maps ℜ m×n × ℜ n×k into ℜ m×k .
Sentence 2: if x and y are column vectors in The identity matrix of arbitrary dimension will be denoted by I.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We identify the resistance diameter of a graph G as R G := max p,q∈V r G (p, q); thus, from the previous theorem, we may also conclude that We complete this section by showing that there exists a family of graphs for which the above inequality is nearly tight.
Sentence 2: we consider the "flower graph" (see Figure 4) obtained by connecting the first vertex of a chain with p − 1 vertices to the root vertex of an m-ary tree of depth one.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, K is the P -invariant Krylov space generated by P and R. An incremental variant of the Krylov-based approach is called Bellman error basis functions (BEBFs) [9].
Sentence 2: given a set of basis functions Φ k (where the first one is assumed to equal R), the next basis is defined to be In the model-free reinforcement learning setting, φ k+1 can be approximated by the temporal-difference (TD) error φ k+1 = r + γ Qk (s , π k (s )) − Qk (s, a), given a set of stored transitions in the form (s, a, r, s ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The intuition behind the algorithm is that since the difference in payoff obtained by using S 2 instead of S 1 is given by (s 2,t − s 1,t ), it is sufficient to emulate Algorithm 1 on this sequence.
Sentence 2: we set x t = t−1 j=1 ρ t−1−j (s 2,j − s 1,j ) and predict ḡ(x t ) (note that since |s 1,t − s 2,t | ≤ 2, we need to use g( 1 2 x) in the definition of ḡ to scale the payoffs).
Relationship: entailment
Domain: NeurIPS
Sentence 1: As in HMC, the idea is to translate the task of sampling from the posterior distribution to simulating from a continuous dynamical system which is used to define a Markov transition kernel.
Sentence 2: over any interval h, the differential equation defines a mapping from the state at time t to the state at time t + h. One can then discuss the evolution of the distribution p(z, t) under the dynamics, as characterized by the Fokker-Planck equation for stochastic dynamics [14] or the Liouville equation for deterministic dynamics [20].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Hyperkernels achieve this by weighting the integrand of ( 5) in relation to what is "on the other side" of the hyperkernel.
Sentence 2: we define convolution hyperkernels by setting g z (x, x ) = r(x, z) r(x , z) in (4) for some r : X × X → R. By (3), the resulting hyperkernel always satisfies the conditions of Definition 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This variational family is indexed by parameters η, which are fit to minimize the KL divergence between the variational and true posteriors.
Sentence 2: we let {q η } be the family of fully factorized distributions where we parameterize the variational posterior with η = {(x u , σ x ), (z u , σ zu ), (ã, σ a ), ( b, σ b )}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let n > m, M ⊂ R N be a Riemannian n-dimensional submanifold.
Sentence 2: then for any (dimensionality reduction) map f : M → R m and any open set W ⊂ M, f cannot achieve both perfect precision and perfect recall on W . In this section, we provide a quantitative analysis for the imperfection of f . we prove an upper bound for the precision of a continuous DR map f (thus f achieves perfect recall).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Similar to AdaBoost, DirectBoost works by sequentially running an iterative greedy coordinate descent algorithm, each time directly minimizing true empirical classification error (1) instead of a weighted empirical classification error in AdaBoost.
Sentence 2: for each iteration, only the parameter of a weak classifier that leads to the most significant true classification error reduction is updated, while the weights of all other weak classifiers are kept unchanged.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If m = 2, we can obtain explicit expressions for the reward envelopes and for the algorithm.
Sentence 2: we have two classifiers, and we assume that the outputs of these classifiers lie in the set ≥ λ such that γ 1 > γ, γ 2 < γ, and λ ≥ 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 3 An Accelerated Solution to the Fenchel Game via Optimism We are going to analyze more closely the use of Algorithm 1, with the help of Theorem 1, to establish a fast method to compute an approximate equilibrium of the Fenchel Game.
Sentence 2: we will establish an approximation factor of O(1/T 2 ) after T iterations, and we recall that this leads to a O(1/T 2 ) algorithm for our primary goal of solving min x∈K f (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Under the stick-breaking construction, two hidden variables for each Markov chain are introduced, representing the transition probabilities between the active and inactive states.
Sentence 2: we define a m = p(s tm = 1|s (t−1)m = 0) as the transition probability from inactive to active, and b m = p(s tm = 1|s (t−1)m = 1) as the selftransition probability of the active state of the m-th chain.
Relationship: entailment
Domain: NeurIPS
Sentence 1: N (0, 1)), we obtain which tends to be much simpler to solve than (2) as the first two terms are separable in the components of x.
Sentence 2: for the choices of P mentioned above, we obtain closed form solutions: where + denotes the positive part and |η (s) | is the sth largest entry in η in absolute magnitude and γ * = min{γ ≥ 0 : the estimators are hardrespectively soft-thresholded versions of η j = A ⊤ j y/m which are essentially equal to the univariate (or marginal) regression coefficients θ j = A ⊤ j y/ A j 2 2 in the sense that η j = θ j (1 + O P (m −1 )), j ∈ [n], hence the term "marginal regression".
Relationship: entailment
Domain: NeurIPS
Sentence 1: In general, this neighborhood contains points from M l as well as other manifolds.
Sentence 2: we assume that for all i there exists ≥ 0 such that the nonzero entries of the sparsest solution of corresponds to the d l + 1 neighbors of x i from M l . among all affine subspaces spanned by subsets of the points {x j } j∈Ni and passing near x i up to error, the one of lowest dimension has dimension d l and it is spanned by the d l + 1 neighbors of x i from M l . In the limiting case of densely sampled data, this affine subspace coincides with the d l -dimensional tangent space of M l at x i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: After constructing a convolutional deep belief network, we perform (approximate) inference of the whole network in a feedforward (bottom-up) manner.
Sentence 2: letting I(h k ij ) b k + ( W k * v) ij , we can infer the pooling unit activations as a softmax function: Given a set of poorly aligned face images, our goal is to iteratively transform each image to reduce the total entropy over the pooling layer outputs of a CDBN applied to each of the images.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At a high level, our algorithm works by first approximating the response of an LDS by an autoregressive model of order (𝜏, 𝜏 ), then refining the approximation using wave-filters with a phase component.
Sentence 2: the blocks of 𝑀 and 𝑁 corresponding to filter index ℎ and phase index 𝑝 specify the linear dependence of 𝑦 𝑡 on a certain convolution of the input time series, whose kernel is the pointwise product of 𝜑 ℎ and a sinusoid with period 𝑊/𝑝.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the first example, most of the components of this vector are zero, but the first and the last few elements follow a discrete convex trend.
Sentence 2: we choose β * = (5 2 , 4 2 , 3 2 , 2 2 , 1, 0, . . . , 0, 1, 2 2 , 3 2 , 4 2 , 5 2 ) ∈ R 100 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In fact, the additive noise model assumes that all categories of the variables are placed in the "right" order; furthermore, if X → Y holds according to the additive noise model, then for any observation (x, y), there exists a function g(X) such that the noise E = y − g(x) is independent of X, i.e., it has the same distribution for different values of x.
Sentence 2: the conditional distribution P (Y |X = x) always has the same shape for different values of x, after being properly shifted according to g(x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, it estimates the inverse Hadamard transform of the compressed and noisy signal of each user in I ,j .
Sentence 2: for each i ∈ I ,j , it computes y i W ri,c which can be described as a multiplication between y i e ri (where e ri is the indicator vector with 1 at the r i -th position) and the scaled Hadamard matrix W, followed by selecting the c-th entry of the resulting vector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In general, the estimation problem with two different distributions Pr(x, y) and Pr ′ (x, y) is unsolvable, as the two terms could be arbitrarily far apart.
Sentence 2: for arbitrary Pr(y|x) and Pr ′ (y|x), there is no way we could infer a good estimator based on the training sample.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Clearly, the accuracy of TrackIPTree is a tremendous improvement on both sets of apriori fixed clusters -for any choice of coverage on legitimate IPs, the accuracy of spam IPs by TrackIPTree is far higher than the apriori fixed clusters, even by as much as a factor of 2.5.
Sentence 2: note that when the coverage required on legitimate IPs is 95%, TrackIPTree achieves 95% accuracy in classifying spam on both data sets, compared to the 35 − 45% achieved by the other clusters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For any Each observed variable x ∈ V obs is modeled as random vector in R d , and each hidden variable h ∈ V hid as a random vector in R k . The joint distribution over all the variables V T := V obs ∪ V hid is assumed satisfy conditional independence properties specified by the tree structure over the variables.
Sentence 2: for any disjoint subsets The class of models considered are specified by the following structural and distributional assumptions.
Relationship: entailment