sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: . . , g), for any g ∈ H. , and b = U 2 2η are constants.
Sentence 2: if Essentially the same theorem holds for SILK, but now with a slightly larger constant a = 4 η(1 In addition, denote g * the minimizer of the batch learning cumulative risk t R(x t , y t , g), and f * the minimizer of the minimum expected risk with R(f * ) := min f E (x,y)∼P (x,y) R(x, y, f ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: [2016] and is presented in Appendix D. Our first lemma, proved in Appendix B.1, confirms that if h✓ h0i , ✓ ⇤ i > 0, then h✓ hti , ✓ ⇤ i > 0 for every t and w hti 1 2 (0.5, 1).
Sentence 2: the estimates of the Population EM 2 remain in the correct hyperplane, and the weight moves in the right direction, too.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Many previous works focused on the settings when only unlabeled data from the target domain are available [Huang et al., 2006, Sugiyama et al., 2008, Yu and Szepesvári, 2012.
Sentence 2: a line of research has been established based on distribution discrepancy, a loss induced metric for the source and target distributions [Mansour et al., 2009, Ben-David et al., 2007, Blitzer et al., 2008, Cortes and Mohri, 2011, Mohri and Medina, 2012.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , W d ∈ R n d−1 ×n d , in which n 0 = n and n d = c. Discussions about networks with more advanced architectures like convolutions, pooling and skip connections can be directly generalized [1].
Sentence 2: we have in which w k = W d [:, k] and σ is the nonlinear activation function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [12,13,14,15,16] and references therein) and the phenomenology, that we shall review briefly here, is well known.
Sentence 2: for β small enough, the global minimum of the Bethe free energy corresponds to the so-called paramagnetic state As we increase β, above a certain value β R , the model enters a retrieval phase, where the free energy has local minima correlated with the factors X and Y .
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, there exists a simple reparameterization of the spike and slab prior that is more amenable to approximate inference methods.
Sentence 2: assume a Gaussian random variable w qm ∼ N ( w qm |0, σ 2 w ) and a Bernoulli random variable s qm ∼ π sqm (1 − π) 1−sqm .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For our validation score, we choose a function of the form: where g is an L-Lipschitz loss function.
Sentence 2: the logistic loss and the hinge loss are 1-Lipschitz, whereas the 0/1 loss is not L-Lipschitz for any L. Other examples of 1-Lipschitz but non-convex losses include the ramp loss: g(w, x, y) = min(1, max(0, 1 − yw x)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We initially compute R(  We split the computation of the inverse centrality R(i) into two terms, namely R(i) = T (i) + S(i), where T (i) and S(i) are the sum of the resistances of vertex i to each descendant and nondescendant, respectively.
Sentence 2: we compute κ(i) and T (i), i = 1, . . . , n with the following leaves-to-root recursions by computing κ(1) then T (1) and caching the intermediate values.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each video is represented using three modalities, i.e., static appearance, motion, and auditory.
Sentence 2: three visual and audio local keypoint features are extracted for each video: scaleinvariant feature transform (SIFT) [18], spatial-temporal interest points (STIP) [19] and melfrequency cepstral coefficients (MFCC) [20].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using ACNN to incorporate the side information can improve the performance more significantly.
Sentence 2: for ACNN 2-conv, the error rate decreases 0.94% (11% relatively) from 8.49% to 7.55%, while the error rate decreases 0.65% (18% relatively) from 3.57% to 2.92% for ACNN 4-conv.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Second, in terms of classification, the DualSGD-Hinge and DualSGD-Logit outperform other methods for almost all datasets except the poker data.
Sentence 2: the DualSGD-based methods achieve the best mistake rates 4.83±0.21, 8.35±0.20, 19.28±0.00 for the cod-rna, ijcnn1 and airlines data, that are, respectively, 32.4%, 11.3%, 8.8% lower than the error rates of the second best modelstwo recent approaches FOGD and NOGD.
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, with the proper rescaling we show that �R N ,w (x), R N ,w (x � )� ≈ κ S (x, x � ).
Sentence 2: then, we also show that the functions obtained by composing bounded linear functions with R N ,w are approximately the bounded-norm functions in H S . the functions expressed by N under varying the weights of the final layer are approximately bounded-norm functions in H S . For simplicity, we restrict the analysis to the case k = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In kernel PCA, the encoding linear operator applies to the s-th tensor power x ⌦s of the data.
Sentence 2: the encoding and decoding are parameterized by a linear operator A 2 R k⇥d s , H pca k,s = (h A , g A ) : h A (x) = Ax ⌦s , , g A (y) = A † y , where A † denotes the pseudo-inverse of A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance.
Sentence 2: we derive sublinear (O(1/t)) convergence on general smooth and convex objectives, and linear convergence (O(e −t )) on strongly convex objectives, in both cases for general sets of atoms.
Relationship: entailment
Domain: NeurIPS
Sentence 1: : inc(X) ≤ K max{r,log n} n . Lemma 3.3.
Sentence 2: we have  a choice of ν 3 = (6inc3(X)) p (2deg 3 (Y )) 1−p , ν 1 = (6inc1(X)) p (2deg 1 (Y )) 1−p for any p ∈ [0, 1] is a valid choice that guarantees exact recovery.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A second synthetic dataset consists of a full matrix of pairwise comparisons derived from a given ranking of n items, with added uncertainty for items which are sufficiently close in the true ranking of items.
Sentence 2: given a positive integer m, we let C In Figure 2, we measure the Kendall ⌧ correlation coefficient between the true ranking and the retrieved ranking, when varying either the percentage of corrupted comparisons or the percentage of missing comparisons.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The observation process N is a marked point process [8] defined on [0, ∞) × R m , meaning that each point, representing the firing of a neuron, is identified by its time t ∈ [0, ∞), and a mark θ ∈ R m . In this work the mark is interpreted as a parameter of the firing neuron, which we refer to as the neuron's preferred stimulus.
Sentence 2: a neuron with parameter θ is taken to have firing rate in response to state x, where H ∈ R m×n and Σ tc ∈ R m×m , m ≤ n, are fixed matrices, and the notation y 2 M denotes y T M y.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We address this challenge by developing a dual formulation for (5).
Sentence 2: given a convex and differentiable function (z), we can rewrite it in its convex conjugate form as (z) = max α∈Ω αz − * (α) , where * (α) is the convex conjugate of (z) and Ω is the domain of dual variable [4].
Relationship: entailment
Domain: NeurIPS
Sentence 1: A set X ⊂ M, where is called geodesically convex if any two points of X are joined by a geodesic lying in X .
Sentence 2: if x, y ∈ X , then there exists a path γ : [0, 1] → X such that γ(0) = x and γ(1) = y.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, it is unclear whether a universal kernel can be constructed from persistence landscapes in a way similar to the definition of k U σ .
Sentence 2: we are not aware of a proof that the construction of persistence landscapes, considered as functions in L 2 , is continuous with respect to d W,q for some 1 ≤ q ≤ ∞.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since r < δ Hp = H p (δ up ) ≤ δ up , by Lemma 3.1 we have, for some M up > 0.
Sentence 2: now, since r and r − M up r 3 are both monotonic for 0 ≤ r ≤ δ up , we have (see H(r, δ up ) ≤ r < δ Hp = H p (δ up ), i.e, H(r, δ up ) < H p (δ up ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This allows us to perform Rao-Blackwellization, conditioning on the samples at the neighbors of x s rather than using x s 's samples directly.
Sentence 2: using this trick we can often get much higher quality estimates from the inference for small N . if the variable state spaces are sufficiently small that they can be discretized (for example, in 3 or fewer dimensions the discretized domain size d may be manageable) but the resulting factor domain size, d b , is intractably large, we can evaluate (4) on the discretized grid for only O(dN b−1 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the HDP-SLDS, we additionally sample R. Sampling z 1:T As shown in [5], the mixing rate of the Gibbs sampler for the HDP-HMM can be dramatically improved by using a truncated approximation to the HDP, such as the weak limit approximation, and jointly sampling the mode sequence using a variant of the forward-backward algorithm.
Sentence 2: we compute backward messages m t+1,t (z t ) ∝ p(y t+1:T |z t , y t−r+1:t , π, θ) and then recursively sample each z t conditioned on z t−1 from where ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To verify that the improved results for NLTop are not the result of an increased number of parameters, we additionally trained another DeepStruct model containing more parameters, which is called DeepStruct++ in Table 2.
Sentence 2: we fixed the original DeepStruct potentials and learned two additional 2-layer multilayer perceptrons that further transformed the unary and pairwise potentials.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To make use of the fingerprinting lemma, we consider a variant of Algorithm 2 that does not truncate the quantity a j − p j to the range ±2 when computing the score z j i for each element i.
Sentence 2: we consider scores based on the quantities We prove two main statements: first, that these untruncated scores are equal to the truncated ones with high probability as long as the mechanism's answers are accurate.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If W has covariance matrix Σ, then (W, W x) has covariance , and V ar(W |W x) = Σ − Σxx T Σ x T Σx whenever x T Σx = 0 [15].
Sentence 2: the conditional variance is the Schur complement of the vector variance x T Σx in the full covariance matrix V . By substituting sample covariance matrices for their population counterparts, we arrive at a new deflation technique: Schur complement deflation Schur complement deflation, like projection deflation, preserves positive-semidefiniteness.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to ensure the identifiability of the model in KCDC, we need to ensure that we apply the same kernel to the response variable when computing the KCDC measures S X→Y and S Y →X .
Sentence 2: we fix a kernel k r with some fixed hyperparameters and apply it both to Y when computing S X→Y and to X when computing S Y →X .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The key idea behind using Lyapunov functions to show stability of the system (1) is similar to that of gradient descent on strictly quasiconvex functions: if one can show that, given a policy π, applying the dynamics f on the state maps it to strictly smaller values on the Lyapunov function ('going downhill'), then the state eventually converges to the equilibrium point at the origin (minimum).
Sentence 2: the assumptions in Theorem 1 below imply that v is strictly quasiconvex within the region of attraction if the dynamics are Lipschitz continuous.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Weight perturbation operates by explicit exploration of the entire M N -dimensional weight space; only one component of a particular update will be in the direction of the true gradient for a given input, while the other components can be viewed as noise masking that signal.
Sentence 2: an update can be written as ∆W = ∆W (the 'learning signal', the actual gradient) + (∆W − ∆W ) (the 'learning noise'), where the average is taken over all values of ψ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let T be the corresponding "true" transition operator, which maps the t-th sample X to the t + 1-th in that chain.
Sentence 2: t (X t |X t−1 ) = P(X t | X)C( X|X t−1 )d X. T produces P(X) as asymptotic marginal distribution over X (as we consider more samples from the chain) simply because P(X) is the marginal distribution of the joint P(X)C( X|X) to which the chain converges.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The algorithms in [8,10,11] can also be considered variants of SPA, with different deflation procedures and pre-/post-processing.
Sentence 2: the algorithm in [8] avoids normalization -for real-word data, normalization at the factorization stage may amplify noise and damage the good conditioning of the data matrix brought by pre-processing, e.g., the tf-idf procedure [8].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note, that M ij = x i • e j , where e j is the j-th unit vector.
Sentence 2: the j-th target is a linear function that picks the j-th column out of M. It is important to understand that the matrix M, which we call the problem matrix, specifies n learning problems: In the jth problem each of the n instances (rows) are labeled by the jth target (column).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unfortunately, this exponential mapping would distort the relationship between perturbations in the latent state (whose size is set by the covariance matrix Q) and the resulting fluctuations in firing rates.
Sentence 2: the size of firing-rate fluctations would grow exponentially with the mean, an effect that would then add to the usual linear increase in spike-count variance that comes from the Poisson output distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We say that an algorithm A has β-approximate max-information of k if for every distribution S over n-element datasets, I β ∞ (S; A(S)) ≤ k, where S is a dataset chosen randomly according to S. We denote this by An immediate corollary of our definition of approximate max-information is that it controls the probability of "bad events" that can happen as a result of the dependence of A(S) on S. Theorem 4.
Sentence 2: let S be a random dataset in X n and A be an algorithm with range Y such that for some β ≥ 0, P[(S, We remark that mutual information between S and A(S) would not suffice for ensuring that bad events happen with tiny probability.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Without loss of generality, we can assume that w (a) is such that i w (a) i = 1 for all a, 1 ≤ a ≤ r. Given this normalization, RANKCENTRALITY estimates w (a) as stationary distribution of an appropriate Markov chain on G. The transition probabilities are 0 for all (i, j) / ∈ E. For (i, j) ∈ E, they are function of Pa .
Sentence 2: transition matrix p(a) = [p Finally, p(a) i ] be a stationary distribution of the Markov chain defined by p(a) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: For the vector case, this property represents a sufficient condition such that log(R) can be defined as a real-valued matrix.
Sentence 2: under property (P1), there exists a (not necessarily unique 2 ) matrix log(R) ∈ M d (R) whose matrix exponential e log(R) = R; cf., Culver [1966], Higham [2014].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, expected error of the coded linear inverse is strictly less than that of uncoded: where J 2 and J 4 are the submatrices of FΛF := J 1 J 2 J 2 J 4 n×n and the matrix Λ is defined in (8).
Sentence 2: (J 1 ) k×k is GΛG , (J 2 ) k×(n−k) is GΛH , and Consider an alternative way of doing linear inverse using n > k workers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we present an end-to-end generative model that jointly synthesizes 3D shapes and 2D images via a disentangled object representation.
Sentence 2: we decompose our image generation model into three conditionally independent factors: shape, viewpoint, and texture, borrowing ideas from classic graphics rendering engines [Kajiya, 1986].
Relationship: entailment
Domain: NeurIPS
Sentence 1: It suffices to set γ t = q(Y )/p(Y ), where p is the distribution of Y t and q is any distribution with mean X * .
Sentence 2: the choice Ỹt := γ t (x t )x t x t with γ t (x) = N (x|0, X * )/N (x|0, Σ t ) satisfies this.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Draw two independent samples Z 1 , Z  [Rakhlin et al., 2015] as well as their data-dependent counterparts [Rakhlin et al., 2011].
Sentence 2: the final ingredient needed for expressing our learning guarantees is the notion of discrepancy between target distribution and the distribution of the sample: The discrepancy is a natural measure of the non-stationarity of the stochastic process Z with respect to both the loss function L and the hypothesis set H. note that if the process Z is i.i.d., then we simply have = 0 provided that q t s form a probability distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the optimal solution for the unconstrained problem has 2 norm no more than 1/λ (see the full version for a proof), we will focus on optimizing θ over the constrained set C = {a ∈ R p | a 2 ≤ 1/λ}, which will be useful for bounding the 1 sensitivity of the empirical loss.
Sentence 2: before we formally introduce the covariance perturbation algorithm due to [19], observe that the optimal solution θ * can be computed as θ * only depends on the private data through X y and X X.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While the motivating optimization problem (Q) is similar to that considered in the COSSO (Lin and Zhang, 2006) for smoothing splines, the SpAM backfitting algorithm decouples smoothing and sparsity, through a combination of soft-thresholding and smoothing.
Sentence 2: spAM backfitting can be carried out with any nonparametric smoother; it is not restricted to splines.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that n 0 is exactly the minimum number of samples required for structured signal recovery from linear measurements when using convex regularizers [3,1].
Sentence 2: the optimization problem succeeds at recovering an unknown weight vector w * with high probability from n observations of the form y i = ⟨a i , w * ⟩ if and only if n ≥ n 0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a first step we wish to find its projection P x (Z) onto the tangent space.
Sentence 2: we wish to find the three matrices M , N 1 and N 2 such that P projecting onto A's columns, denoted P A , is exactly equal to AA † We can similarly define P A ⊥ ,P B and P B ⊥ .
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, we note that it is easy to extend CRH to learn nonlinear hash functions via the kernel trick [26].
Sentence 2: according to the generalized representer theorem [27], we can represent the projection vectors w x and w y as where φ x (•) and φ y (•) are kernel-induced feature maps for modalities X and Y, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unlike the   In a final experiment we tested the ability of DS-PSRL algorithm in continuous state and action domains.
Sentence 2: we implemented the discrete infinite horizon linear quadratic (LQ) problem in Szepesvári [2015, 2011]: x t+1 = A ⇤ x t + B ⇤ u t + w t+1 and c t = x T t Qx t + u T t Ru t , where t = 0, 1, ..., u t 2 R d is the control at time t, x t 2 R n is the state at time t, c t 2 R is the cost at time t, w t+1 is the 'noise', A ⇤ 2 R n⇥n and B ⇤ 2 R n⇥d are unknown matrices while Q 2 R n⇥n and R 2 R d⇥d are known (positive definite) matrices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Define then we have Since we only have finite samples, the expectations in Proposition 1 have to be approximated by empirical moments.
Sentence 2: they are computed by averaging over indices j = 1, 2, . . . , n. For each permutation (a, b, c) ∈ {(2, 3, 1), (3, 1, 2), (1, 2, 3)}, we compute The statement of Proposition 1 suggests that we can recover the columns of C c and the diagonal entries of W by operating on the moments M 2 and M 3 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work we investigate compound HD (hierarchical-deep) architectures that integrate these deep models with structured hierarchical Bayesian models.
Sentence 2: we show how we can learn a hierarchical Dirichlet process (HDP) prior over the activities of the top-level features in a Deep Boltzmann Machine (DBM), coming to represent both a layered hierarchy of increasingly abstract features, and a tree-structured hierarchy of classes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More precisely, we set T = u t+1 q ⊗ u q ⊗ u q + =q σ u ⊗ u ⊗ u in (4) and then find optimal u t+1 q by minimizing the least squares objective given by (4).
Sentence 2: each inner iteration is a simple least squares problem over the known entries, hence can be implemented efficiently and is also embarrassingly parallel.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper we take a different approach in order to jointly learn the processed fluorescence signal and the inverse covariance estimate.
Sentence 2: we convolve the fluorescence time series f i with a kernel α and then pass the convolution through the logistic function σ(x), Note for α 0 = −α 1 (and α k = 0 otherwise) this convolution filter approximates the threshold filter in Equation 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This turns out to be fundamentally different to our noisy setting.
Sentence 2: the results of [8] state that exact recovery is impossible, and even with approximate recovery, a huge number of tests (i.e., higher than polynomial) is needed unless = O q , where q max is the maximum allowed reconstruction error measured by the Hamming distance, and is maximum adversarial noise amplitude.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given how well EXP does as an approximate recognition model for DBM, and EXP does not bother to detect change points or modify its learning rate in response to detected change points, there might exist a generative model for which EXP would be an exact Bayesian recognition model.
Sentence 2: one might consider a model that assumes the underlying real-valued hidden variable to undergo persistent stochastic changes with constant noisy characteristics, such as a Gaussian process, which then gives rise to noisy binary or categorical observations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In fact, if ω * = 0, and if Assumption A is not fulfilled, then 3 ∃ g ∈ ∆P * such that A T g = 0.
Sentence 2: given the matrices A and C, there are two distinct vectors θ 1 and θ 2 in ∂P such that A T θ 1 = A T θ 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Supplementary Material goes in detail about the preprocessing, including R/MATLAB scripts to generate the data.
Sentence 2: the we generated data from "interventions" simulated from a model that was fitted to an actual study on premature infant development [3], where the dose is the number of days that an infant is assigned to follow a development program and the outcome is an IQ test at age 3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 The functional relationship between ✓ and its joint counterpart e ✓ = e ✓ i,pa(i) (x) is mediated through a kernel-dependent conditioning operation ⌧  ( e ✓) = ⌧  ( e w, e µ µ µ, e B) = ✓ and can be computed in closedform for a wide range of kernels, including Gaussian, cosine, logistic and other kernels with sigmoid CDF.
Sentence 2: for block decompositions e , we obtain y 0 (y 0 e µ (j) See Appendix A.1 for a detailed derivation.
Relationship: entailment