sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: (3) since by design s T D G 1 = 0.)
Sentence 2: to find the correct setting of γ 1 , it suffices to perform a binary search over the possible values of γ 1 in the interval (−vol(G), λ 2 (G)) until the correlation constraint is satisfied, that is, until (s T D G x) 2 is sufficiently close to κ 2 1 , see [8,14].
Relationship: entailment
Domain: NeurIPS
Sentence 1: To learn a metric which approximates the ideal geometric setup described above, we introduce, for each training point, a conditional distribution over other points (as in [5]).
Sentence 2: 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.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Other works have studied the risk landscapes of neural network optimizations for more complex structures under various model assumptions [9,10,11,12,13,14,15,16,17,18,19].
Sentence 2: references [13,14,15] consider a two-layer neural network with Gaussian inputs under a matched (realizable) model where the output is generated from a network with planted weights.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Such implicit bias, which can also be viewed as a form of regularization, can play an important role in learning.
Sentence 2: implicit regularization has been shown to play a crucial role in training deep models [14,13,18,11]: deep models often generalize well even when trained purely by minimizing the training error without any explicit regularization, and when there are more parameters than samples and the optimization problem is underdetermined.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Analytic differentiation of optimization Finally, our work relates methodologically to a recent trend of incorporating more structured layers within deep networks.
Sentence 2: recent work has looked incorporating quadratic programs [Amos and Kolter, 2017], combinatorial optimization [Djolonga and Krause, 2017], computing equilibria in games [Ling et al., 2018], or dynamic programming [Mensch and Blondel, 2018].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The trades that achieve this optimal drop in risk are called efficient: Our following key result shows that efficient trades have remarkable structure: once the state r and subset S is specified, there is a unique efficient trade, up to cash transfers.
Sentence 2: the surplus is removed from the position vectors and then redistributed as cash to the traders; the choice of trade is merely in how this redistribution takes place.
Relationship: entailment
Domain: NeurIPS
Sentence 1: That is, any continuous Markov process (with no jumps) with the desired invariant distribution can be cast within our framework, including HMC, Riemann manifold HMC, SGLD, SGHMC, their recent variants, and any future developments in this area.
Sentence 2: our method provides a unifying framework of past algorithms, as well as a practical tool for devising new samplers and testing the correctness of proposed samplers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To be able to exploit this, we denote M = { mi } M i=1 as the set of possible values for illuminant chromaticity m, and construct it from a training set.
Sentence 2: we quantize 1 the chromaticity vectors { mt } T t=1 of the illuminants in the training set, and let M be the set of unique chromaticity values.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that F t i is submodular and that if F t (S) ≥ 1 then F t i (S) ≥ 1.
Sentence 2: in this sense F t i is an easier objective than F t . Also, for any v, the gain of F t i at ∅ is the normalized gain of F t at S i−1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , the m th according to π k−m+1 and then starts again.
Sentence 2: formally, this can be written as It is straightforward to show that the value v π k,m of this periodic non-stationary policy π k,m is the unique fixed point of the following operator: Finally, it will be convenient to introduce the following discounted kernel: for any pair of values v and v , it can easily be seen that T 3 Tightness of the performance bound of Theorem 1 The bound of Theorem 1 is tight for API in the sense that there exists an MDP [4,Example 6.4] for which the bound is reached.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The next result shows that, when the number of voters tends to infinity (and the weight of each voter tends to zero), the variance of W Q will tend to 0 provided that the average of the covariance of the risks of all pairs of distinct voters is ≤ 0.
Sentence 2: the variance will always tend to 0 if the risk of the voters are pairwise independent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper presents a probabilistic-graphical model that can be used to infer characteristics of instantaneous brain activity by jointly analyzing spatial and temporal dependencies observed in electroencephalograms (EEG).
Sentence 2: we describe a factor-graph-based model with customized factor-functions defined based on domain knowledge, to infer pathologic brain activity with the goal of identifying seizure-generating brain regions in epilepsy patients.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, we focus on sets of size ≤ m/ 3 (at most 1/ 3 elements for each function) to find an initial set such that the remaining elements have marginal value ≤ 3 V i for f i , for every i.
Sentence 2: one can try all possible initial sets of this size (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The main feature of proposed algorithms is summarized as follows: Our algorithms achieve a sample complexity of O(1/ε) with much better dependency on the problem dimension than the ones in existing work.
Sentence 2: if the smallest magnitude of the non-zero components of the optimal solution is not too small, the rate can be boosted to near the minimax optimal sample complexity of full information algorithms.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , y m } ⊆ Y two sets of observations between which we would like to find a correspondence.
Sentence 2: we would like to find some permutation π ∈ Π m on m terms, that is m×m and π1 m = 1 m and π such that the pairs Z(π) := (x i , y π(i) ) for 1 ≤ i ≤ m correspond to dependent random variables.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For local constant approximation, the error term α x−x ′ is the first order in x−x ′ ; for local linear approximation, the error term β x−x ′ 2 is the second order in x − x ′ ; for local quadratic approximation, the error term ν x − x ′ 3 is the third order in x − x ′ .
Sentence 2: if f (x) is smooth with relatively small α, β, ν, the error term becomes smaller (locally when x − x ′ is small) if we use a higher order approximation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The ergodic decomposition theorem for stationary processes (see, e.g., 7) states that any stationary process can be expressed as a mixture of stationary ergodic processes.
Sentence 2: a stationary process ρ can be envisaged as first selecting a stationary ergodic distribution according to some measure W ρ over the set of all such distributions, and then using this ergodic distribution to generate the sequence.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Described this way, it is clear that the Plackett-Luce model is basically a partial size-biased permutation of the atoms in G [10], and the existing machinery of random measures and exchangeable random partitions [11] can be brought to bear on our problem.
Sentence 2: in Section 2 we will use a gamma process as the prior over the atomic measure G. This is a completely random measure [12] with gamma marginals, such that the corresponding normalized probability measure is a Dirichlet process.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For this, we recall a closely related result shown by de Farias and Van Roy [1] for the ALP.
Sentence 2: de Farias and Van Roy [1] showed that given an appropriate weighting (or in their context, 'Lyapunov') function ψ, one may solve an ALP, with ψ in the span of the basis functions Φ; the solution to such an ALP then satisfies: provided β(ψ) ≤ 1/α.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Throughout the execution, we use the inline comments in order to prescribe a feasible (but not necessarily optimal) assignment of the data points to (a subset of k of) the current candidate centers.
Sentence 2: we maintain an array ASSIGN, where we write ASSIGN[j] = u (for some center u in our current set of candidate centers) to denote that all of the points in the optimal cluster S * j are assigned to the candidate center u.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will also show that the capsule projection can be viewed as normalizing the multiple columns of the weight matrix simultaneously to form an orthogonal basis, which makes it more effective in incorporating novel components of input features to update capsule representations.
Sentence 2: the capsule projection can be viewed as a multi-dimensional weight normalization in capsule subspaces, where the conventional weight normalization is simply a special case of the capsule projection onto 1D lines.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then with probability at least 1 − δ, C is (σ, ǫ) consistent.
Sentence 2: the clusters containing A[X] and A ′ [X] are internally connected and mutually disconnected in C(r) for r defined by where ρ is now slightly modified (in constants), i.e., ρ := min σ 7 , ǫτ 72d , τ 24 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assumptions 3 and 4 are the key technical conditions under which we can generalize the analyses in [2,3,4] to the general GLM case.
Sentence 2: we can show that the statements of the following propositions hold, which show that the random vectors X following the GLM graphical model in (10) are suitably well-behaved: Proposition 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Viewing E = A -A as a perturbation to the "ideal" A that results in A = A+E, we ask: When can we expect the resulting rows of Y to cluster similarly to the rows of Y?
Sentence 2: when will the eigenvectors of L, which we now view as a perturbed version of L, be "close" to those of L?
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here, 𝐸 𝑖 and 𝐸 𝑖+1 are the system energies at solution configuration 𝑖 and 𝑖 + 1 respectively; 𝐾 is the Boltzmann constant; and 𝑇 is the system temperature.
Sentence 2: a step will be taken when a lower energy is found.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 7 For a generic kernel a0, the shift-coherence is bounded by | a0, sτ [a0] | ≈ 1/ √ k for any shift τ .
Sentence 2: 8 under a Bernoulli-Gaussian model, for each j, E[z 2 j ] = θ i =0 a0, si[a0] 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the same spirit, we investigate the classification calibration property of the AT k loss.
Sentence 2: we first obtain the population form of the AT k objective using the infinite limit of (2) We then consider the optimization problem where the infimum is taken over all measurable function f : X → R. We say the AT k (aggregate) loss is classification calibrated with respect to p if f * has the same sign as the Bayes rule f c . The following theorem establishes such conditions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since [∆ − ∆ * ] + is a monotonically increasing function of ∆, its upper bound is derived directly from the upper bound of ∆.
Sentence 2: the upper bound of ∆ is given by and the upper bound of the surrogate loss with a fixed ∆ * is given by When the original space is fixed, the problem is not straightforward since the surrogate loss becomes a V-shaped function with ρ > 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we show that several characteristic phenomena of multistability derive naturally from applying standard MCMC inference to Markov random fields (MRFs) -high dimensional, loosely coupled graphical models with spatial structure characteristic of many low-level and mid-level vision problems.
Sentence 2: we capture the classic findings of Gamma-distributed mode-switching times in bistable perception; the biasing effects of contextual stimuli; the situations in which fused (rather than bistable) percepts occur, and the propagation of perceptual switches in traveling waves across the visual field.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the prediction of out-of-sample travel times, ML outperformed both a random model and the constant model used for the initialization (a(x, s, γ) = α(x) with α(x) obtained from (19) with uniform p path ).
Sentence 2: when all journeys in the test dataset are considered, ML outperforms the baseline method with a 24% improvement.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Indeed, we can recover the coupling layer from the autoregressive transformation of MAF by setting µ i = α i = 0 for i ≤ d and making µ i and α i functions of only x 1:d for i > d (for IAF we need to make µ i and α i functions of u 1:d instead for i > d).
Sentence 2: both MAF and IAF can be seen as more flexible (but different) generalizations of Real NVP, where each element is individually scaled and shifted as a function of all previous elements.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Bias and MSE In [17], the bias and MSE of the standard SG-MCMC algorithms with a Kth order integrator were analyzed, where the order of an integrator reflects how accurately an SG-MCMC algorithm approximates the corresponding continuous diffusion.
Sentence 2: if evolving x t with a numerical integrator using discrete time increment h induces an error bounded by O(h K ), the integrator is called a Kth order integrator, e.g., the popular Euler method used in SGLD [3] is a 1st-order integrator.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our first observation is that the collision probability can be well approximated using the χ 2 similarity: Figure 4 shows this approximation is better than 1 π cos −1 (ρ 1 ).
Sentence 2: in sparse data, the approximation is very accurate (except when ρ χ 2 is close to 1), while the bound 1 π cos −1 (ρ 1 ) is not sharp (and the curve is not smooth in ρ 1 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inspired by recent model distillation methods in [20,29,34], we use a smoothed underlying distance matrix to solve the "vanishing gradient" problem when updating B and Λ.
Sentence 2: the 9) is replaced by a Sinkhorn distance with the smoothed underlying distance, i.e., y S (B(R), λ m (A); D τ θ ), where (•) τ , 0 < τ < 1, is an element-wise power function of a matrix.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While the existence of very small adversarial perturbations seems counter-intuitive in this low-dimensional illustrative example (i.e., r in and r unc can be large for some choices of f ), we show in the next sections that this is the case in high dimensions.
Sentence 2: if for all i, P(C i ) ≤ 1 2 (the classes are not too unbalanced), we have To see the dependence on the number of classes more explicitly, consider the setting where the classes are equiprobable, i.e., P(C i ) = 1 K for all i, K ≥ 5, then This theorem is a consequence of the Gaussian isoperimetric inequality first proved in [33] and [34].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper we give necessary and sufficient conditions under which kernels of dot product type k(x, y) = k(x . y) satisfy Mercer's condition and thus may be used in Support Vector Machines (SVM), Regularization Networks (RN) or Gaussian Processes (GP).
Sentence 2: we show that if the kernel is analytic (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While our model is flexible enough to deal with different sets of unlabeled examples in the support set and the query set, for a task T ∼ P (T ) we only use a single unlabeled set of examples U T to follow the same training scheme in [Ren et al., 2018], for a better comparison with our baseline models.
Sentence 2: for MetaGAN with MAML, we set S u T = S s T and Q u T = U T . For MetaGAN with RN, we set S u T = ∅ and Q u T = U T in loss eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However we now give an explicit construction for the minimal method of choosing the weak hypothesis for which the number of iterations of greedy covering and InfoBoost grow linearly in k and the number of iterations of AdaBoost with Bias is quadratic in k. For any dimension N we define an example set which is the rows of the following (N + 1) × N dimensional matrix x: All entries on the main diagonal and above are +1 and the remaining entries −1.
Sentence 2: the last row is all −1 (See Figure 4).
Relationship: entailment
Domain: NeurIPS
Sentence 1: as the problem of discovering the mapping from image I 1 to image I 2 and searching for an image I 4 that has the same relation to image I 3 as I 1 to I 2 .
Sentence 2: we find a function T (parametrized by θ) that maps each pair of images (I 1 , I 2 ) to a vector x 12 = T (X 1 , X 2 ; θ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use this change of variables to define a new function f , parameterized by θ, that is equivalent to the original function f : The gradient and the Hessian of this new function f are (by the chain rule): A gradient descent iteration θt = θt−1 − η∇ f ( θ) for the transformed function corresponds to for the original parameter θ.
Sentence 2: by left-multiplying the original gradient with a positive definite matrix D −1 , we effectively apply gradient descent to the problem after a change of variables θ = D 1 2 θ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, assuming p(p+1) > m, we show that if the search space of (SDP) is compact and if the search space of (P) is a regularly defined smooth manifold, then, for almost all cost matrices C, if Y satisfies first-and second-order necessary optimality conditions for (P), then Y is a global optimum of (P) and, since p(p+1) 2 ≥ m, X = Y Y is a global optimum of (SDP).
Sentence 2: first-and second-order necessary optimality conditions for (P) are also sufficient for global optimality-an unusual theoretical guarantee in non-convex optimization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, we run experiments on both simulation and real-world data and show our method outperforms the standard approaches to this problem.
Sentence 2: we use data from a large-scale RCT measuring the effect of small classrooms and teacher's aids [WJB + 90, Kru99] to obtain ground-truth estimates of causal effects, which we then try and reproduce from a confounded observational study.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this situation, discrimination control and utility are still guaranteed for samples drawn from q D,X,Y that are transformed using p Ŷ , X|X,Y,D , where the latter is obtained by solving (6) with p D,X,Y .
Sentence 2: denoting by q Ŷ |D and q X, Ŷ the corresponding distributions for Ŷ , X and D when q D,X,Y is transformed using p Ŷ , X|X,Y,D , we for n sufficiently large (the distortion control constraints (4) only depend on p Ŷ , X|X,Y,D ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let split k ∈ {1, . . . , p} be the index of the feature that the root node of the k th tree split on and suppose that that split reduces the impurity of the label by r k . Then DStump assigns feature j the score  Choosing d: We pick the number of features to use in our local linear model via a greedy forward selection procedure that relies on the fact that features can be sorted by their s j scores.
Sentence 2: we evaluate the predictive accuracy of MAPLE on a held out validation set for d = 1, . . . , p and choose the value of d that gives the best validation accuracy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We compute the optimal policy using Bellman's dynamic programming principle (Bellman, 1952).
Sentence 2: we iteratively compute the expected cost of continue and stop as a function of the belief state b t (these are the Q-factors for continue and stop, , then the optimal policy chooses to continue; otherwise, it chooses to stop; therefore, the belief state is partitioned by the decision policy into a continuation region and a stopping region (details omitted due to lack of space).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Reeb graph R f (X) provides a simple yet meaningful abstraction of the input domain X w.r.t function f . Computation in discrete setting.
Sentence 2: assume the input domain is modeled by a simplicial complex K. a k-dimensional simplex σ is simply the convex combination of k + 1 independent points {v 0 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This results in asynchronous updates, in which the activity of a neuron x i changes stochastically as a function of its input cue xi , the activity of all other neurons, x \i , and neighbouring synapses, W i,• and W •,i .
Sentence 2: the Gibbs sampler results in a sigmoid transfer function, with the total current to the neuron given by the log-odds ratio: with the terms I in/out rec defining the evidence from the incoming and outgoing synapses of neuron i, and the constants a and b determined by the prior over patterns and the noise model.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let n (i) be the length of document x (i) .
Sentence 2: for any k ∈ {0, 1}, we have: where 1{•} is the indicator function (1{True} = 1, 1{False} = 0), and we have applied Laplace (add-one) smoothing in obtaining the estimates of the word probabilities.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Remarkably, our theory reveals how all these detailed retinal properties arise as a natural consequence of the statistical structure of natural movies and realistic energy constraints.
Sentence 2: our theory simultaneously accounts for: (1) why it is beneficial to have these multiple cell types in the first place, (2) why the four properties of cell density, spatial RF-size, temporal filtering speed, and contrast sensitivity co-vary the way they do across midget and parasol types, and (3) quantitatively explains the variation in midget to parasol density ratios over retinal eccentricities.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This problem is feasible in both λ j s and t and hence we can drop the minimization over individual constraints to obtain an equivalent problem: min λj ∈∆n j ∀j,t t subject to nj k=1 λ jk ≤ t. One can now eliminate t by reintroducing the max j and interchange the min λj ∈∆n j ∀j with other variables to obtain: Now one can derive the standard dual of (3) wrt.
Sentence 2: to the variables w jk , b, ξ i alone, leading to: where α, γ are Lagrange multipliers, The following points regarding (4) must to be noted: • (4) is equivalent to the well-known SVM [18] formulation with kernel 1 γ * j is the weight given to the j th component and λ * jk is weight given to the k th kernel of the j th component.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The key observation of Flaxman et al.
Sentence 2: [11], which is later reused in all of the follow-up work, is that g t can be estimated by randomly perturbing the point x t . on round t, the algorithm chooses the point instead of the original point x t , where > 0 is a parameter that controls the magnitude of the perturbation, A t is a positive definite d ⇥ d matrix, and u t is drawn from the uniform distribution on the unit sphere.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We make our FBN to only depend on text (and not on the image), making its learning task easier.
Sentence 2: our FBN performs the following computation: f sent LSTM, dim 256 f phrase linear+mean pool f f bn 3-layer MLP with dropout +3-way softmax Here, w c t and w f t denote the one-hot encoding of words in the sampled caption and feedback sentence, respectively.
Relationship: entailment