sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: However, we present results for PIM and RIN only, as the results on the other datasets mostly followed the results on these and in a way these two datasets are the most extreme.
Sentence 2: they are the smallest and largest with dimensions 768 ⇥ 8 (768 examples, and 8 dimensional feature space), and 7200 ⇥ 20, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The heuristic part of our strategy is to skip f i (x s ) or f i (x t ) if our evaluation of f i has been zero more than two consecutive times (and skipping it an exponentially larger number of times each time it remains zero).
Sentence 2: for each example i we maintain two variables, sk i (for 'skip') and ps i (for 'pass').
Relationship: entailment
Domain: NeurIPS
Sentence 1: The design philosophy behind RNs is to constrain the functional form of a neural network so that it captures the core common properties of relational reasoning.
Sentence 2: the capacity to compute relations is baked into the RN architecture without needing to be learned, just as the capacity to reason about spatial, translation invariant properties is built-in to CNNs, and the capacity to reason about sequential dependencies is built into recurrent neural networks.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Based on a sufficient condition presented in Section 3.1, we study how "invariance" of the subdifferential of one function would lead to nontrivial proxdecompositions.
Sentence 2: we prove in Section 3.3 that when the subdifferential of one function is scaling invariant, then the prox-decomposition always holds if and only if another function is radial-which is, quite unexpectedly, exactly the same condition proven recently for the validity of the representer theorem in the context of kernel methods [9,10].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore applying Talagrand's inequality in the version of [3] to the class G r we obtain Let us define ε r := 30ϕn(r) r + 2τ v nr 2−ϑ 1/2 + 7τ nr .
Sentence 2: then the above inequality gives with probability not less than 1 − e −τ that for all (p, f ) ∈ P × F we have and consequently we have with probability not less than 1 − e −τ that Now observe that for the functions h 1 and h 2 which we defined when estimating (12) and (13) we have and hence we can combine our estimates (16), (14), and (15) of the terms (11), (12), and (13) to obtain that with probability not less than 1 − 3e −τ we have for r satisfying the assumption (10) we have 30ϕn(r) , and 7τ nr ≤ 1 4 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then the resulting distribution Pr (X), as defined by Equation 1, is a simple-route distribution for graph G. This theorem will not hold if Pr (B) were not a simple-route distribution for graph G B .
Sentence 2: having each distribution Pr (A i | b i ) be a simple-route distribution for graph G bi is not sufficient for the hierarchical distribution to be a simple-route distribution for G. Hierarchical distributions that satisfy the conditions of Theorem 1 will be called hierarchical simpleroute distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The essence of the proposed method is to improve the prediction by i) parametrizing the targets (e.g., 3D boxes) instead of directly estimating the targets, and ii) cooperative training across different modules in contrast to training these modules individually.
Sentence 2: we parametrize the 3D object bounding boxes by the predictions from several modules, i.e., 3D camera pose and object attributes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the positive side, if we restrict ourselves to tree-structured graphs, the estimator ( 6) is attractive, since it relies only on an estimate Σ of the population covariance Σ * that satisfies the deviation condition (7).
Sentence 2: when the samples {x i } n i=1 are contaminated by noise or missing data, all we require is a sufficiently good estimate Σ of Σ * .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In fact a recent paper [11] has shown that if only the distances as encoded by the graphical model depicted in figure 1 are taken into account (nodes represent points in S and states represent points in U), exact probabilistic inference in such a model can solve the isometric problem optimally.
Sentence 2: an energy function of the following form is minimised: In [11], it is shown that loopy belief propagation using this model converges to the optimal assignment, and that the number of iterations required before convergence is small in practice.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Thompson sampling algorithm simply samples actions according to the posterior probability they are optimal.
Sentence 2: actions are chosen randomly at time t according to the sampling distribution π TS t = α t . By definition, this means that for each a ∈ A, P(A t = a|F t−1 ) = P(A * = a|F t−1 ) = α t (a).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also looked at spike rasters from different cell-types to gain insight into their phase relationship with the stimulus.
Sentence 2: we present recordings for the site indicated by the arrow (see Figure 3a) for gratings moving in eight directions ranging from 0 to 360 degrees in 45-degree increments (this location was chosen because it is in the vicinity of a pinwheel, is reasonably selective, and shows considerable modulation in its firing rate).
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , b T a sequence of strategic buyers b1 , .
Sentence 2: . . , bT such that the regret incurred to A by b1:T is at least the switching cost regret incurred to A 0 by b 1:T . The idea behind the construction is as follows: At each iteration t we choose with probability half to present to the seller b t and with probability half we present to the seller a buyer z t that has the following statistics: z t is with probability 1 2 a buyer with value v = 1 2 and patience ⌧ = 0, and with probability 1 2 , z t is a buyer with value v = 1 and patience ⌧ = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In such a case, the resulting algorithm is simply the constant step-size Exponential Weights , and the adaptive step sizes , 1 11 and Furthermore, if only one player (say, Player I) follows the above algorithm, her regret against any sequence x 1 , .
Sentence 2: . . , x T of plays is this implies the worst-case regret of O log(nT ) √ T in the general setting of online linear optimization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We further allow some samples are corrupted and not drawn from (1).
Sentence 2: the set of sample indices {1, 2, . . . , n} is separated into two disjoint subsets: if i-th sample is in the set of "good" samples, which we name G, then it is a genuine sample from (1) with the parameter Θ * .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now return to the setting with inputs and show how the Bayesian strategy for the Gaussian regression model is a minimax strategy for all input sequences x ≤T .
Sentence 2: if we fix the input sequence x ≤T , we can consider the competitor class to be we make the more stringent comparison against a model class which has full knowledge of the input sequence in advance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, we propose to replace the context hyper-parameters with trainable models, that we learn jointly with the embeddings on an objective that preserves the graph structure (the Negative Log Graph Likelihood, NLGL).
Sentence 2: we propose Graph Attention Models, using a softmax to learn a free-form contexts distribution with a parameter for each type of context similarity (e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, by Markov inequality, we have that Pr i∼p (t) (•) err Di (g (t) ) > 12 ≤ .1 .
Sentence 2: now consider the total weight By Lemma 1 and E, we have that Combining ( 2), (3), and (4), we have = k, we have the following inequality holds for every t = 0, 1, 2, . . . : us focus on an arbitrary player i.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Furthermore, the regularizer J(y) can be interpreted as a Reproducing Kernel Hilbert Space (RKHS) squared semi-norm, giving us additional insight into this choice of regularizer: Kp induced by the kernel with a null-space of all constant functions.
Sentence 2: f Kp is the norm of the projection of f onto the RKHS induced by then J(f ) can be written as a squared semi-norm induced by the kernel with a null-space spanned by indicator functions 1 [ai,bi] (x) on the connected components of Ω.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , x p ∈ R n . Further, for a vector v, sign(v) is the vector with entries sign(v An outline of this section is as follows: (1) We first consider the zero-noise problem W = 0, and prove several useful properties of the Lasso estimator in this case.
Sentence 2: we show that there exists a threshold for the regularization parameter below which the support of the Lasso estimator remains the same and contains supp(θ 0 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to alleviate this problem, we propose to update the weight for the existing support vector whose classification confidence is significantly affected by the new misclassified example.
Sentence 2: we consider a support vector (x b , y b ) ∈ D for weight updating if it satisfies the following two conditions "conflicts" with the new misclassified example (x a , y a ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is because the joint estimator leverages the labels on the target items (whose true values are unknown), and relies less on the control items.
Sentence 2: as the number n t of target items increases, the optimal value of k for the joint estimator decreases with a rate of 1 √ n t (see Figure 1(d)), but that of the two-stage estimator stays the same.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given that the numerical task that must be solved in KDR is the same as the one to be solved in kernel ICA, we can import all of the computational techniques developed in [9] for minimizing KGV.
Sentence 2: the optimization routine that we use is gradient descent with a line search, where we exploit incomplete Cholesky decomposition to reduce the n × n matrices to low-rank approximations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , φ(x T , g α )} can be related to coin tosses.
Sentence 2: we associate a coin with each dimension i ∈ {1, 2, . . . , d}, and consider the set of coin bias vectors lying in the set Given a particular function g α ∈ G(δ) (or equivalently, vertex α ∈ V), we consider the oracle φ that presents noisy value and gradient samples from g α according to the following prescription: • Pick an index i t ∈ {1, . . . , d} uniformly at random.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More specifically, we are interested in the ǫ-mixing time τ (ǫ), which guarantees a distribution that is ǫ-close to P L in terms of total variation.
Sentence 2: we must spend at least this many time steps in order to acquire a sample from a distribution close to P L . Our next result states that the chain in Algorithm 1 mixes rapidly:  One advantage of having a rapidly-mixing Markov chain as means of sampling from a DPP is that it is robust to addition/deletion of elements.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Following [3,4], we will approach this problem via neighborhood estimation, where we estimate the neighborhood of each node individually, and then stitch these together to form the global graph estimate.
Sentence 2: if we have an estimate N (s) for the true neighborhood N * (s), then we can estimate the overall graph structure as: In order to estimate the neighborhood of any node, we consider the sparsity constrained conditional MLE.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Even without communication during the game, the planning phase gives the team members an advantage: for instance, the team members could skew their strategies to use certain actions to signal about their state (for example, that they have particular cards).
Sentence 2: by having agreed on each member's planned reaction under any possible circumstance of the game, information can be silently propagated in the clear, by simply observing public information.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In these works, the idea is to select a small subset of features with maximum dependence with the target variable Y .
Sentence 2: the task is to find a subset of variables S ⊆ {1, . . . , d} with |S| ≤ k based on the following optimization problem where X S (X i ) i∈S and I (X S ; Y ) denotes the mutual information between the random variable X S and Y .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this field, a number of studies have focused on constructing better low dimensional representations that combine various features of brain activities [3,4], because the problem of classifying EEG signals is intrinsically high dimensional.
Sentence 2: efforts have been made to reduce the number of electrodes by eliminating electrodes recursively [8] or by decomposition techniques e.g., ICA, which only uses the marginal distribution, or Common Spatial Patterns (CSP) [9] which additionally takes the labels into account.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our first result in this paper provides sufficient condition for W = −B ⊗ A to be concave.
Sentence 2: it provides a connection between conditionally positive (or negative) definite functions (Wendland, 2004), and negative definiteness of −B ⊗ A: Φ is conditionally positive definite of order 1 if for all pairwise distinct points Conditionally negative definiteness is defined analogously.
Relationship: entailment
Domain: NeurIPS
Sentence 1: There has been a large amount of work on differential privacy for a wide range of statistical and machine learning tasks over the last decade [6,30,13,21,33,24,1]; for overviews, see [18] and [29].
Sentence 2: algorithms for the private maximization problem (and variants) have been used as subroutines in many applications; examples include PAC learning [25], principle component analysis [14], performance validation [12], and multiple hypothesis testing [32].
Relationship: entailment
Domain: NeurIPS
Sentence 1: To circumvent these difficulty, we devised a modified algorithm called PMED-Hinge (Algorithm 1) that involves extra exploration.
Sentence 2: we modify the optimization problem as follows: Let R H (1),...,(L) ({µ i,l }, {θ i }, {κ l }, {δ i,l }) = {q i,l } ∈ Q : inf {θ i }∈T c (1),...,(L) ,{κ l }∈K all :∀ l∈[L] dKL(µ (l),l ,θ (l) κ l )≤δ i,l (i,l)∈[K]×[L]:i =(l) q i,l (d KL (µ i,l , θ i κ l ) − δ i,l ) + ≥ 1 , where (x) + = max(x, 0).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, to enable GAIL to produce diverse solutions, we condition the discriminator on the embeddings generated by the VAE encoder and integrate out the GAIL objective with respect to the variational posterior q (z|x 1:T ).
Sentence 2: we train the discriminator by optimizing the following objective A related work [20] introduces a conditional GAIL objective to learn controllers for multiple behaviors from state trajectories, but the discriminator conditions on an annotated class label, as in conditional GANs [21].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We address this notoriously hard challenge, under the assumptions that the function varies only along some low-dimensional subspace and is smooth (i.e., it has a low norm in a Reproducible Kernel Hilbert Space).
Sentence 2: we present the SI-BO algorithm, which leverages recent low-rank matrix recovery techniques to learn the underlying subspace of the unknown function and applies Gaussian Process Upper Confidence sampling for optimization of the function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: There exists G > 0 such that max 3.
Sentence 2: the distribution µ is full-dimensional: there exists C µ > 0 such that for all x ∈ X and the next lemma states that, under Assumption 1, G n is a consistent estimate of G. Lemma 4 ([14, Theorem 1]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: For basal trees of CA3 pyramidal cells, this analysis yielded an estimated "corrected" value of b between 0.14 and 0.25, with a remaining much smaller than b. Interestingly, our analysis based on extrapolation shows that, regardless of the assumed amount of distortion present in the experimental data, given the numbers measured for CA3 basal trees, positive initial <α> implies positive initial b.
Sentence 2: not only measurement errors, but also possible biological distortions of the dendritic tree may not be capable of accounting for the observed positivity of the parameter b.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The kernel is usually computed from the adjacency matrices A 1 and A 2 of the two graphs, but it must be invariant to the ordering of the vertices.
Sentence 2: writing the kernel explicitly in terms of A 1 and A 2 , we must have k(A 1 , A 2 ) = k(A 1 ,P A 2 P ) for any permutation matrix P . Permutation invariance has proved to be the central constraint around which much of the graph kernels literature is organized, effectively stipulating that graph kernels must be built out of graph invariants.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is easy to see that both fully connected and convolutional networks of any depth L can realize any linear predictor β ∈ R D . The model class expressed by both networks is therefore simply the unconstrained class of linear predictors, and the two architectures are merely different (over) parameterizations of this class Thus, the empirical risk minimization problem in (4) is equivalent to the following optimization over the linear predictors β = P(w): (5) Although the optimization problems ( 4) and ( 5) are exactly equivalent in terms of the set of global minima, in this paper, we show that optimizing (4) with different parameterizations leads to very different classifiers compared to optimizing (5) directly.
Sentence 2: consider problems (4)/(5) on a linearly separable dataset {x n , y n } N n=1 and using the logistic loss (the two class version of the cross entropy loss typically used in deep learning).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Sparse penalized regression models are analyzed with the purpose of finding, to the extent possible, regimes of dictionary invariant performance.
Sentence 2: a Type II Bayesian estimator with a dictionarydependent sparsity penalty is shown to have a number of desirable invariance properties leading to provable advantages over more conventional penalties such as the ℓ 1 norm, especially in areas where existing theoretical recovery guarantees no longer hold.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a given instantiation m i , let Σ(m i ) denote the covariance matrix implied by the model and λ k (m i ) be the value of coefficient λ k . Definition 1.
Sentence 2: a coefficient, λ k , is identified if for any two instantiations of the model, m i and m j , we have λ k is identified if it can be uniquely determined from the covariance matrix, Σ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we leverage unbiased gradient estimators based on (8) in conjunction with our optimal degree distribution (11) to design computationally efficient methods for solving (4).
Sentence 2: we propose to randomly sample a degree n from (11) and estimate the gradient via Monte-Carlo method: where ∂w (k) j /∂θ i can be computed using a Rademacher vector v (k) and the recursive relation (6).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In doing so, we first sample a variable u 1 that indicates which source to sample from.
Sentence 2: when u 1 = −1, u 1 = 0, or u 1 = l > 0, we respectively sample θ 1 from p ν , p qµ , or T (φ l , •).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Another question we can ask is what happens if we allow a larger rank than the rank of the optimal solution.
Sentence 2: if we have f (X ⇤ ) = 0 with low rank(X ⇤ ), even rank(X ⇤ ) = 1, but consider (1) or (2) with a high r. Could we still have non-global local minima?
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assume that V Y X , V Y Z , and V ZX are Hilbert-Schmidt, and that the regularization constant ε n satisfies ε n → 0 and ε 3 n n → ∞.
Sentence 2: then, we have the convergence in probability ÎNOCCO n → I N OCCO and ÎCOND n → I CON D (n → ∞) in probability.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The variance of the plug-in estimators Ĝk , Gk is given by In the above expressions, c 1,i , c 1 , c 2 , c 4 and c 5 are constants that depend only on g, f and their partial derivatives, and I = {1, . . . , d}.
Sentence 2: the constants c 1,i , c 1 , c 2 , c 4 and c 5 are independent of k, N and M . From Theorem 1, k → ∞ and k/M → 0 for the estimators Ĝk and Gk to be unbiased.
Relationship: entailment
Domain: NeurIPS
Sentence 1: S by π S such that π S (i) = d i /d(S) if i ∈ S and π S (i) = 0 otherwise.
Sentence 2: in terms of the conductance of S, the following theorem gives an upper bound on the expected probability mass escaped from S if the distribution of the starting vertex is π S . Theorem 3.1 shows that most of the probability mass will be absorbed in S, provided that S is of small conductance and the random walk starts from S according to π S . a PARW will be trapped inside the cluster 1 from where it starts, as desired.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2, and indicate that the early layers of VGG16 (in particular, Front and Layer3) are better predictors of human sensitivity than the deeper layers (Layer4, Layer5, Layer6).
Sentence 2: the most noticeable eigen-distortions from representations within VGG16 become more discriminable with depth, but so generally do the least-noticeable eigen-distortions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Variational approximation with Contrastive Divergence: Since the data likelihood L(Θ) is generally intractable to compute, our method is based on the efficient contrastive divergence technique [16,25,26,28].
Sentence 2: we derive a variational approximation L v (q 0 , q 1 ) of the negative log-likelihood L(Θ) , that is: L v (q 0 , q 1 ) := R(q 0 (x, z, h), p(x, z, h)) − R(q 1 (x, z, h), p(x, z, h)), where R(q, p) is the relative entropy, and q 0 is a variational distribution with x and z clamped to their observed values while q 1 is a distribution with all variables free.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This reduces the problem to showing that each term in the sum of ( 6) corresponds to the contribution of all prefixes of x li : vJ Assume we descend down the path xli : Vi] in S(y) (e.g., for the string bab with respect to the tree of Figure 2 this would correspond to (root, b, bab», then each of the prefixes t along the path (e.g., (' , , b, ba, bab) for the example tree) occurs exactly as many times as Ivs( ceil( t)) does.
Sentence 2: prefixes ending on the same edge occur the same number of times.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The set W i→i can be partitioned according to the number of times that walks return to node i: i→i is the set of all self-return walks which return to i exactly r times.
Sentence 2: w A walk which starts at node i and returns r times is a concatenation of r single-revisit self-return walks, so ρ(W i→i ) r . This means: This geometric series converges since the model is walk-summable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: One possible construction algorithm is an extension of Kruskal's algorithm for the MST where we pick edges without replacement and restart for every new tree.
Sentence 2: we sort the list of N (N −1) 2 edges e ij by increasing distance d ij and visit each available edge in turn, removing it if it merges two clusters (or equivalently does not create a cycle); whenever we have removed N − 1 edges, we go back to the start of the list.
Relationship: entailment