sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: With ω t ∼ N (0, H t ) as an estimate of the system noise, C can be decomposed as C t = A t Σ t A T t + H t . Note that while the transition dynamics in our generative model operates on the inferred latent space, it takes untransformed controls into account.
Sentence 2: we aim to learn a latent space such that the transition dynamics in z linearizes the non-linear observed dynamics in x and is locally linear in the applied controls u. Reconstruction of an image from z t is performed by passing the sample through multiple hidden layers of a decoding neural network which computes the mean p t of the generative Bernoulli distribution 1 P θ (X|Z) as where h dec θ (z t ) ∈ R n d is the response of the last hidden layer in the decoding network.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To realize this idea, we advocate a novel input layer for deep neural networks that takes a topological signature (in our case, a persistence diagram), and computes a parametrized projection that can be learned during network training.
Sentence 2: this layer is designed such that its output is stable with respect to the 1-Wasserstein distance (similar to [27] or [1]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In online isotonic regression, the natural goal is to predict the incoming data points as well as the best isotonic function in hindsight.
Sentence 2: for time steps t = 1, . . . , T , the learner observes an instance x i ∈ R, makes a prediction y i of the true label y i , which is assumed to lie in [0, 1].
Relationship: entailment
Domain: NeurIPS
Sentence 1: • When all the functions are self-concordant, we establish a similar guarantee if both the gradient and Hessian of the function can be queried multiple times.
Sentence 2: we propose to apply the damped Newton method multiple times in each round, and prove an O(min(P * T , S * T )) bound of the dynamic regret under appropriate conditions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The main contributions of this paper lie in the theoretical analysis of the proposed method.
Sentence 2: we show: 1) the proposed method can improve the estimation bound of the standard Dantzig selector approximately from Cs 1/p √ log mσ to C(s − N ) 1/p √ log mσ where the value N depends on the number of large entries in β * ; 2) when N = s, the proposed algorithm can achieve the oracle solution with a high probability; 3) with a high probability, the proposed method can select the same number of correct features under a milder condition than the standard Dantzig selector method.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given parameter n ∈ N and sample access to a distribution over N, there exists an algorithm (Algorithm 1) for T(PBD n , ) which takes 2 samples, and runs in time Note that the sample complexity of our algorithm is n 1/4 • Õ(1/ 2 ), matching the informationtheoretic lower bound up to a logarithmic factor in 1/ .
Sentence 2: our algorithm does not incur the extraneous Ω(1/ 6 ) term of [1].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The key construction is a variational representation of M-estimation that can express a number of standard robust (and non-robust) methods in a common framework.
Sentence 2: consider the following adaptive form of loss function where r is a residual value, is a closed convex base loss, η is an adaptive weight on the base loss, and ψ is a convex auxiliary function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As the true label signature can be a tiny fraction of a given sequential data point, we break the data point itself into multiple overlapping instances such that at least one of the instance cover the true label signature (see Figure 1).
Sentence 2: given a sequential data point X i , we construct a bag represents the τ -th instance (or sub-window) of data points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, as We can now derive the dual of ESP-design (we consider only which admits as dual We easily show that H ⪰ 0 and that g reaches its maximum on S ++ m for A such that ∇g = 0.
Sentence 2: h and A are co-diagonalizable, with Λ Diag(e ℓ−1 (Λ (i) ))Λ = Diag(h 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We show that achieving the guarantee of (1) significantly more efficiently than the work of [MM16,MW17] is likely very difficult.
Sentence 2: we prove that for a wide class of kernels, the kernel low-rank approximation problem is as hard as multiplying the input A 2 R n⇥d by an arbitrary C 2 R d⇥k .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let k and n be positive integers.
Sentence 2: for each k-bit string y := y 1 • • • y k , let ⇢ y be an nqubit mixed state such that for each i 2 {1, 2, . . . , k}, there is a two-outcome measurement E 0 that depends only on i and the prefix v := y 1 y 2 • • • y i 1 , and has the following properties (iii) if y i = 0 then Tr(E 0 ⇢ y )  a v ", and (iv) if y i = 1 then Tr(E 0 ⇢ y ) a v + ", where " 2 (0, 1/2] and a v 2 [0, 1] is a "pivot point" associated with the prefix v. Then k = O n/" 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This example is adapted from [12] and [18], which considers the problem of power dispatch.
Sentence 2: at each time step t, we try to minimize the power generation cost c i (x t,i ) for each generator i while maintaining the power balance n i=1 x t,i = d t , where d t is the power demand at time t. Also, each power generator produces an emission level E i (x t,i ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, since the joint probability of (X, Y ) is unknown but we are provided with an independent and identically distributed (iid) sample of observations {(x i , y i )} n i=1 , we resort to minimizing the empirical risk: , within a class H ⊂ (R) X of functions, calibrated in order to overcome the shift from the true risk to the empirical one.
Sentence 2: when H has the form: R being a convex function and c > 0 a constant, Takeuchi et al.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our experiments suggest that both steps of Autowarp -first, learning latent representations using sequence-to-sequence autoencoders, and second, finding a warping distance that agrees with the latent representation -are important to learning a good similarity metric.
Sentence 2: we carried out experiments with deeper autoencoders to determine if increasing the capacity of the autoencoders would allow the autoencoder alone to learn a similarity metric.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Steps (2) and (3) of the method update x, decreasing the energy (1) by first taking a gradient descent step with respect to the inner product term in (1) and then taking a "backward" or proximal step involving f . In steps (4) and ( 5), the energy (1) is increased by first marching up the gradient of the inner product term with respect to y, and then a backward step is taken with respect to g. PDHG has been analyzed in the case of constant stepsizes, ⌧ k = ⌧ and k = .
Sentence 2: it is known to converge as long as ⌧ < 1/⇢(A T A) [4,5,11].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Core to our work is a new and arguably more natural analysis of the Sinkhorn iteration algorithm, which we show converges in a number of iterations independent of the dimension n of the matrix to balance.
Sentence 2: this analysis directly suggests a greedy variant of Sinkhorn iteration that also provably runs in near-linear time and significantly outperforms the classical algorithm in practice.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The above refinement is obtained by mimicking the analysis of Hedge forecasters for small losses (see, e.g., [3,Section 2.4]).
Sentence 2: one should substitute Lemma 1 with the following lemma in the analysis carried out in Section 5; its proof follows from the mere replacement of Hoeffding's inequality by [3,Lemma A.3], which states that for all η ∈ R and for all random variable X taking values in [0, 1], one has ln E[e −ηX ] (e −η − 1)EX.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At each iteration l, our best approximation of π is given by the pushforward density p l = (T l • • • • • T 1 ) * p 0 .
Sentence 2: the SVGD algorithm then seeks a transport map T l+1 = I + Q that further decreases the KL divergence between (T l+1 ) * p l and π, for an appropriate choice of Q ∈ H d . the SVGD algorithm seeks a map Q ∈ H d such that where 0(x) = 0 denotes the zero map.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The optimal solution to the LP: max 3 The Known Objective Problem In this section, we focus on the Known Objective Problem where the coefficients of the objective function c are fixed and known to the learner but the feasible region P (t) on day t is unknown and changing.
Sentence 2: p (t) is the intersection of a fixed and unknown polytope .
Relationship: entailment
Domain: NeurIPS
Sentence 1: By combining Theorem 2.1 with previous results [3,4,5], we immediately obtain bounds on the accuracy of the matrix Dantzig selector (4) and the matrix Lasso (5).
Sentence 2: for the first time we can show universal recovery of low-rank matrices via Pauli measurements, and near-optimal bounds on the accuracy of the reconstruction when the data is noisy [5].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since SNIPER is trained on 512x512 size chips, it can reap the benefits of a large batch size and training with batch-normalization on a single GPU node.
Sentence 2: we can use a batch size of 20 per GPU (leading to a total batch size of 160), even with a ResNet-101 based Faster-RCNN detector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Separate motion systems have been proposed to explain human perception in scenarios like this example.
Sentence 2: braddick [1] proposed that there is a short-range motion system which is responsible for perceiving movements with relatively small displacements (e.g., the player's movement), and a long-range motion system which perceives motion with large displacements (e.g., the flying baseball), which is sometimes called apparent motion.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The relation between walks and the geometric series (10) is that the entries of R l correspond to walk-sums over l-length walks from i to j in the graph, i.e., (R l ) i,j = ρ(W i l →j ).
Sentence 2: hence, the variance σ 2 i ≡ P i,i of variable i is the walk-sum taken over the set W i→i of self-return walks that begin and end at i (defined so that (i) ∈ W i→i ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: As noted by [16], people are capable of recognizing that individual stimuli possess an arbitrarily large number of characteristics, but in any particular context will make judgments using only a finite, usually small number of properties that form part of our current mental representation.
Sentence 2: by moving to a Bayesian nonparametric form, we are able to bring the ADCLUS model closer to the kinds of assumptions that are made by psychological theories.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , z n ) denotes the Cartesian coordinates on R n and d n z = dz 1 . . . dz n denotes the volume form on R n . In particular, Before proving this theorem, we prove some results on the relation between u p (q) and d p (q).
Sentence 2: there exist δ up > 0 and M up > 0 such that for all q with d p (q) ≤ δ up , we have, lim q→p up(q) dp(q) = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The distributed coding scheme has important consequences for interpreting neural responses: since knowledge about the underlying distribution is spread across the population, the activity of single cells does not reflect the underlying computation in any obvious way.
Sentence 2: although the network did reproduce various properties of single neuron and pairs of neuron responses seen experimentally, we found that their modulation with uncertainty provides relatively limited information about the underlying probabilistic computation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, even though L[x, y] itself is learned through knowledge of co-occurence of chromaticities in natural images, estimation of the illuminant during inference is still achieved through a simple aggregation of per-pixel distributions.
Sentence 2: we set the entries of L[x, y] to minimize a cost function C over a set of training images: where mt is the true illuminant chromaticity of the t th training image, and p t i is computed from the observed colors v t (n) using (4).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Qualitatively quite different from both epitomes and previous stel models, the stel epitome is a model flexible enough to be applied to a very diverse set of images.
Sentence 2: we are interested in datasets that might represent well a human's total visual input over a longer period of time, and so we captured two weeks worth of SenseCam images, taken at a frequency of roughly one image every 20 seconds during all waking hours of a human subject over a period of two weeks (www.research.microsoft.com/∼jojic/aihs).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the completeness case, we claim that these steps will reject with at least 99% probability unless F r (p) = O(C (n 1−r + m −r )).
Sentence 2: if F r (p) ≥ KC (n 1−r + m −r ), then m p r ≥ 1, and therefore, by Claim 2.4 we have that E[ γ r ] = F r (p) and Var( γ r ) = O(F r (p) 2 /K 2 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we report our results using the general bound ) where ↵ can be any constant form the interval [0.5, 1].
Sentence 2: the observation that the optimal values of the expected loss and empirical loss are within a V n distance of each other implies that there is no gain in improving the optimization error of minimizing L n beyond the constant V n . if we find an approximate solution w n such that the optimization error is bounded by , then finding a more accurate solution to reduce the optimization error is not beneficial since the overall error, i.e., the sum of estimation and optimization errors, does not become smaller than V n . Throughout the paper we say that w n solves the ERM problem in (2) to within its statistical accuracy if it satisfies We can further leverage the estimation error to add a regularization term of the form (cV n /2)kwk 2 to the empirical loss to ensure that the problem is strongly convex.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Any two nodes i and j are connected (independent of other edges) with probability p if they belong to the same cluster and with probability q otherwise.
Sentence 2: pr(ll|ll) = p, Pr(lm|ll) = 1 − p, Pr(ll|lm) = q and Pr(lm|lm) = 1 − q.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To prove convergence bounds for stochastic-gradient based methods, we usually require b g(w) to be an unbiased estimator of the full-batch gradient, and to have small variance Ekb g(w)k 2 , as this controls the speed of convergence.
Sentence 2: to see this, suppose w ⇤ is a critical point of f , then we have the progress made by the algorithm at a single step is, in expectation, controlled by the term E[kb g(w k )k] 2 2 ; the smaller it is, the bigger the progress.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Because Q( θ k+1 ; θ k ) involves terms log A k i,j and log π k i , in practice we bound A k i,j and π k i away from zero to ensure that Q does not blow up.
Sentence 2: we assume a small positive constant θ min so that A k i,j ≥ θ min and π k i ≥ θ min .
Relationship: entailment
Domain: NeurIPS
Sentence 1: From ( 6), it is easy to see that the (E,E)-block of the Hessian is a diagonal matrix given by ( Using this diagonal block, we erase (V,E)-block and (E,V)-block of the Hessian.
Sentence 2: we choose a square matrix X such that det X = 1 and After the computation given in the supplementary material, we see that From , it is easy to check that I N + D − Â = Y W , where Â and D is defined in (9) and W is a diagonal matrix defined by For the left equality, theorem 2 is used.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our goal is to compare the fidelity of different samples approximating a common target distribution.
Sentence 2: we seek to quantify the discrepancy between E Q and E P in a manner that (i) detects when a sequence of samples is converging to the target, (ii) detects when a sequence of samples is not converging to the target, and (iii) is computationally feasible.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that unlike the previous models in [9][10][11][12][13][14][15][16], with the exception of [17], a policy in our model is context-dependent.
Sentence 2: since the state is (t, µ t , X(P π t )) and not just the time-belief tuple (t, µ t ), a policy π can recommend different actions for the same belief and at the same time but for a different context.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We propose a simple baseline which selects harder examples with higher probability, as done by Loshchilov and Hutter [29].
Sentence 2: we let is the history of prediction probability which stores all p(y i |x i ) when x i is selected to train the network before the current iteration t, H = i H t−1 i , pH t−1 i (y i |x i ) is the average probability of classifying sample i into its correct class y i over all the stored p(y i |x i ) in H t−1 i , and D is a smoothness constant.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It has been shown in a variety of literatures [27,11,22,12,24] that the PCA conducted on elliptical distributions shares a number of good properties enjoyed by the PCA conducted on the Gaussian distribution.
Sentence 2: [11] show that with regard to a range of hypothesis relevant to PCA, tests based on a multivariate Gaussian assumption have the identical power for all elliptical distributions even without second moments.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The data, eighty sequences in all, were synthesized from the generating profile sequence of length fifty shown on the top line of the figure.
Sentence 2: each data sequence was created by extracting one to four (mean two) patches from the generating sequence of length three to thirty (mean sixteen), sampling from these patches to produce corresponding patches of amino acids in the data sequence, and then filling in the gaps in the data sequence with amino acids sampled from a uniform distribution over amino acids.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We simulate two data sets, SIM-1 and SIM-2, from ground-truth regression coefficients.
Sentence 2: we simulate a network formation process starting from time unit 0 until time 1200, where nodes arrive in the network at a constant rate λ 0 = 10 (i.e., on average, 10 nodes join the network at each time unit); the resulting simulated networks have 11,997 nodes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Notice that in many, and perhaps even most, practical scenarios, being a better predictor on more instances is more important than avoiding such sporadic "embarrassing predictions".
Sentence 2: our approach fits any commerce and advertising setting where the agent offers predictions to users on the value of different goods or services, aiming at maximizing the number of users that will find her predictions more accurate than the one provided by the expert.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In Section 3, we show that the hypothesis learned with the Empirical Risk Minimization (ERM) procedure based on the local minimax risk closely achieves the optimal local minimax risk.
Sentence 2: we provide a data-dependent bound on the generalization error, which behaves like the bound for ordinary ERM in the no-ambiguity regime (Theorem 1), and excess risk bounds under uniform smoothness assumptions on F (Theorem 2) and a less restrictive assumption that F contains at least one smooth hypothesis (Theorem 3).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The mergeon encodes the height at which clusters A , B, and C merge.
Sentence 2: the fact that M = λ 2 everywhere on A × B represents the merging of A and B at level λ 2 in W. It is clear that in general there is no unique mergeon representing a graphon cluster tree, however, the above definition implies that two mergeons representing the same cluster tree are equal almost everywhere.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Adaptive sampler for MGP The above Gibbs sampler needs a predefined truncation level K c . In [3,26] the authors proposed an adaptive sampler, tuning K c as the sampler progresses, with convergence of the chain guaranteed [16].
Sentence 2: the adaptation procedure is triggered with probability p(t) = exp(z 0 + z 1 t) at the tth iteration, with z 0 , z 1 chosen so that adaptation occurs frequently at the beginning of the chain but decreases exponentially fast.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Therefore, the cost function (3) will heavily penalize the classifier if it predicts an output that is believed to be less probable by the prior distribution p LM (•), and it will not penalize the classifier when it does not predict an output that p LM (•) believes to be probable.
Sentence 2: the classifier is encouraged to predict a single output mode with high probability in p LM (•), a behavior called "mode-seeking" in [25].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We selected i ∈ {6, 14} for grid and pedigree, and i ∈ {2, 4} for protein.
Sentence 2: i = 6 (grids, pedigrees) and i = 2 (proteins) were the smallest i-bounds for which SPRBFAOO could solve at least two thirds of instances within the 2 hour time limit, while i = 14 (grids, pedigrees) and i = 4 (proteins) were the largest possible i-bounds for which we could compile the heuristics without running out of memory on all instances.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The function F ( ) is said to satisfy the local error bound with parameter 2 (0, 1] if 9 > 0 such that for any 2 S , dist( , where C is a positive constant possibly depending on .
Sentence 2: when = 1/2, F ( ) is said to be locally quadratic and when = 1, it is said to be locally linear.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Next, we rewrite the density into the form required by (1) by substituting σ(ψ k ) for π k : Choosing a k (x) = x k and b k (x) = N k for each k = 1, 2, . . . , K − 1, we can then introduce Pólyagamma auxiliary variables ω k corresponding to each coordinate ψ k ; dropping terms that do not depend on ψ and completing the square yields where Ω ≡ diag(ω) and κ(x) ≡ x − N (x)/2.
Sentence 2: conditioned on ω, the likelihood of ψ under the augmented multinomial model is proportional to a diagonal Gaussian distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The learning process of student model then switches to: Such a sequential procedure of obtaining f ω * (i.e., f ω T ) is the learning process of the student model with training data D train and loss function provided via the teacher model µ θ , and we use an abstract operator F to denote it: Just as the training and testing setup in typical machine learning scenarios, the teacher model here similarly follows the two phases setup.
Sentence 2: in the training process of teacher model, similar to qualified human teachers are good at improving the quality of exams, the teacher model in L2T-DLF refines the loss function it sets up via optimizing its own θ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is now easy to dominate P ∈ H f (ρ; c, k) ∪ H f (ρ; c, k) far away from B 0 .
Sentence 2: choose any p 0 ∈ B 0 ∩ P , which was guaranteed to exist in the preceding point; since min p∈P B f (x, p) ≤ B f (x, p 0 ) holds for all x, it suffices to dominate p 0 .
Relationship: entailment