sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: R n . We introduce an empirical risk formulation that has favorable global geometry for gradient methods, as soon as m = O(kd 2 log n), under the model of a d-layer fully-connected neural network with random weights.
Sentence 2: when suitable deterministic conditions on the generator and measurement matrix are met, we construct a descent direction for any point outside of a small neighborhood around the true k-dimensional latent code and a negative multiple thereof.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The class of functions F is a class of submodular functions by Lemma 10 (in Appendix C).
Sentence 2: we now show that every function in F is efficiently learnable from samples drawn from any distribution D. we show that for any ✏, 2 (0, 1) the functions are (✏, ) PAC learnable with the absolute loss function (or any Lipschitz loss function) using poly(1/✏, 1/ , n) samples and running time.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Evaluation Module (E-Module) is used to determine whether a phrase is a complete caption.
Sentence 2: given an input phrase P , the E-Module encodes it into a vector z e , using a two-level LSTM model as described above, and then evaluates the probability of P being a complete caption as It's worth noting that other properties could also be checked by the E-Module besides the completeness.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The moment-to-moment amplitude fluctuations of these local rhythms reflect variable functional states of the underlying neuronal cortical networks and can be used for brain-computer interfacing.
Sentence 2: the pericentral µand β rythms are diminished, or even almost completely blocked, by movements of the somatotopically corresponding body part, independent of their active, passive or reflexive origin.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The third algorithm, which we call H-RLS (Hierarchical Regularized Least Squares), is a simplified variant of APPROX-H-BAYES in which the thresholds τ i,t are set to zero.
Sentence 2: we have g i,t (x t ) = {w i,t x t ≥ 0} where the weights w i,t are defined as in (5) and updated as in the APPROX-H-BAYES algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We begin demonstrating the power of asymmetry by considering an asymmetric variant of the above problem.
Sentence 2: even if S is symmetric, we allow associating with each object x i two distinct binary codewords, u i ∈ {±1} k and v i ∈ {±1} k (we can think of this as having two arbitrary mappings u i = f (x i ) and v i = g(x i )), such that S ij = sign( u i , v j − θ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: If the lower bound behaves roughly like W 2 (P U , P f −1 (V ) ), our simulation result also serves as an empirical evidence that W 2 (P U , P f −1 (V ) ) weighs more on high precision.
Sentence 2: we generate 10000 uniformly distributed samples in a 10-dimensional unit 2 -ball.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Interpreting Z 0 = (X 0 , y 0 ) as the empty set and the associated Tikhonov matrix as C ,0 = I, all the results stated above can be extended to the case r = 0.
Sentence 2: we define the algorithm as w h (Z 0 ) = argmin w2R d w h 2 = h and its transfer risk as E 0 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We establish that for deep linear networks, there is a direct connection between the smoothness parameter of the training loss function and the largest singular value of the estimated linear function.
Sentence 2: we show that if the gradient descent algorithm can converge to a solution with a large step size, the function estimated by the network must have small singular values, and hence, the estimated function must have a small Lipschitz constant.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Algorithm 1 Metropolis-within-Gibbs sampler for missing data imputation using a trained VAE Inputs: Observed data x obs , trained VAE (f θ , g γ ), number of iterations T Outputs: Markov chain of imputations xmiss 1 , ..., xmiss T . At each step of the chain, rather than generating codes according to the approximate posterior distribution, we may use this approximation as a proposal within a Metropolis-Hastings algorithm (Metropolis et al., 1953;Hastings, 1970), using the fact that we have access to the unnormalised posterior density of the latent codes.
Sentence 2: at each step, we will generate a new code zt as a proposal using the approximate posterior Ψ(z|g γ (x obs , xmiss t−1 )).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Tsallis entropy is in fact a generalization of the Shannon entropy, as one obtains the latter as a special case of the former asymptotically.
Sentence 2: it is easy to prove the following uniform convergence: We emphasize again that one can easily show that Tsallis-smoothing bandit algorithm is indeed identical to INF using the appropriate parameter mapping, although our analysis is simpler due to the notion of differential consistency (Definition 2.2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Classical estimators from this field can be classified into two categories: either (i) they are computationally efficient but incur an error that scales polynomially with the dimension d, or (ii) they are provably robust (in the aforementioned sense) but are hard to compute.
Sentence 2: essentially all known estimators in robust statistics (e.g., the Tukey median [Tuk75]) have been shown [JP78,Ber06,HM13] to be intractable in the high-dimensional setting.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, say we have a learning algorithm that assumes everything in the same cluster has the same label (so the hypothesis space H consists of all rules that do not split clusters).
Sentence 2: then if the graph of which clusters are associated with which is an expander graph, then the distributions will be expanding with respect to H. given a labeled example x, the learning algorithm will generalize to x's entire cluster C i , then this will be propagated over to nodes in the associated clusters C j in X 2 , and so on.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is natural to restrict to nonlinear mappings that act radially, preserving the spherical symmetry.
Sentence 2: one can show that the generating function of p(x wht ) is completely determined by its radial marginal distribution: p r (r) = r d−1 β f (−r 2 /2), where r = x wht , Γ(•) is the standard Gamma function, and β is the normalizing constant that ensures that the density integrates to one.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Following [3], there is a natural way to define "features" in a Reeb graph and measure their "importance".
Sentence 2: given a function f : X → IR, imagine we plot its Reeb graph R f (X) such that the height of each point z ∈ R f (X) is the function value of all those points in X mapped to z.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The following proposition extends the result for the one-dimensional total variation [4,23] to all submodular functions and their Lovász extensions: Proposition 5 (Proximal problem for ℓ 1 -penalized problems) The unique minimizer of 1 2 w − z 2 2 + f (w) + λ w 1 may be obtained by soft-thresholding the minimizers of 1 2 w − z 2 2 + f (w).
Sentence 2: the proximal operator for f + λ • 1 is equal to the composition of the proximal operator for f and the one for λ • 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: • Graphs with Random Sparsity Structures: We use the procedure mentioned in Example 1 in [11] to generate inverse covariance matrices with random non-zero patterns.
Sentence 2: we first generate a sparse matrix U with nonzero elements equal to ±1, set Σ −1 to be U T U and then add a diagonal term to ensure Σ −1 is positive definite.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This oracle policy takes as input a sequence of predicted labels ŷ<t , the input x, and the free label multiset with respect to its predictions, Y t = Y\ {ŷ <t }.
Sentence 2: it outputs a distribution whose entire probability (1) is evenly distributed over all the items in the free label multiset Y t . Definition 2 (Oracle).
Relationship: entailment
Domain: NeurIPS
Sentence 1: From ( 10) and ( 13), the θ-th power of the Korobov kernel k α is given by If αθ ∈ N, this is exactly the kernel k αθ of the Korobov space W αθ Kor ([0, 1]) of lower order αθ.
Sentence 2: w αθ Kor ([0, 1]) is nothing but the θ-power of W α Kor ([0, 1]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Without loss of generality, the intercept is set to zero by centering the response and standardizing the predictors as To reduce the complexity of the model and leverage the structural information in X m , we assume that the coefficient tensor W to be both low-rank and sparse.
Sentence 2: we assume W can be decomposed via CP decomposition as r , where each rank-1 tensor W r is possibly sparse, or equivalently, the vectors in its representation w (n) r , n = 1, . . . , N , are possibly sparse, for all r = 1, . . . , R. Here we impose sparsity on the rank-1 components from CP decomposition -rather than on W itself [Chen et al., 2012, Tan et al., 2012.
Relationship: entailment
Domain: NeurIPS
Sentence 1: From Equation 6, if we just use Q ||| instead of Q, we get This equation shows that the unknowns in matrix Q ||| can be found by exploiting the fact that R i is a truncated rotation matrix (as was done in [1]).
Sentence 2: if Λ2i−1:2i denotes the two rows of matrix Λ at positions 2i − 1 and 2i, then we have where I 2×2 is an identity matrix, giving three indepedent constraints for each image i.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The result is generally biased, however [7] have shown that this algorithm offers a very advantageous bias-variance trade-off in case of manifolds with noise.
Sentence 2: we use the output Ŷ of PRINCIPALCURVES to find a subset of points that (1) lie in a high density region relative to most directions in R D and (2) are "in the middle" of their neighbors, or more formally, have neighborhoods of dimension at least d. this is a good heuristic to avoid "border effects", or other regions where the d-manifold assumption is violated.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The noisy data are described by the random variable X + W = Y ∼ ν, and , where Γ is the gamma function.
Sentence 2: ρ2 (µ, ν) ≤ σ √ D. The magnitude of the bound, and hence the worst-case sensitivity of the estimation procedure to noise, is controlled by the noise variance, by the extrinsic dimension, and by the distortion exponent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that this is just a very simple one-dimensional example, but the outlined approach works in any dimension and for any convex objective function, even if we just have access to stochastic subgradients [Orabona and Pal, 2016].
Sentence 2: if the gradients are bounded in a range, the same reduction works using a continuous coin.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In many natural machine learning settings, one is not only faced with data that may be corrupted or deficient in some way (classification noise or other label errors, missing attributes, and so on), but with data that is not uniformly corrupted.
Sentence 2: we might be presented with data of variable quality -perhaps some small amount of entirely "clean" data, another amount of slightly corrupted data, yet more that is significantly corrupted, and so on.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This remark allows us to work with small matrices (of at most d × d instead of r × r) and to avoid the problem of estimating d, the intrinsic dimension of the data manifold.
Sentence 2: we evaluate the tangent subspace around each sampled point x i using weighted (local) Principal Component Analysis (wPCA) and then express g R r | TpM directly in the resulting lowdimensional subspace as the unit matrix I d . The tangent subspace also serves to define a local coordinate chart, which is passed as input to Algorithm 1 which computes H(x i ), G(x i ) in these coordinates.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Of course, in practice, we never perform Gibbs sampling on a variational distribution but instead on the true posterior distribution which is proportional to e f (x) (where we assumed that − log q v (x) is not part of f (x)).
Sentence 2: at each Gibbs step we simulate a new value for some x i from the posterior conditional distribution that is proportional to e f (x \i ,xi) and where x \i are the fixed values for the remaining random variables.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our starting point is a model which regresses a response variable y ∈ R to observed covariates (predictor variables) x ∈ R D by a non-linear functional relationship.
Sentence 2: we consider a multi-linear structure to account for the interactions among the covariates in the model: (1) Here, w i for i = 0, . . . , D are bias and linear weights as in the standard linear regression model, J is the number of multi-way interactions where Z j , β j for j = 1, . . . , J represent the interactions, i.e., sets of indices of interacting covariates and the corresponding interaction weights, respectively.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since these messages function as any other network activation, gradients can be passed back along the channel, allowing end-to-end backpropagation across agents.
Sentence 2: the network, which we call a C-Net, outputs two distinct types of values, as shown in Figure 1(b), a) Q(•), the Q-values for the environment actions, which are fed to the action selector, and b) m a t , the real-valued vector message to other agents, which bypasses the action selector and is instead processed by the discretise/regularise unit (DRU(m a t )).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that solving this problem from observed data on users as we described is made challenging both by the fact that unobserved variables causally affect both community membership and sharing, creating spurious correlations, and because social network membership introduces dependence among users.
Sentence 2: for realistic social networks, every user's activity potentially depends on every other user's activity (even if indirectly).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The negative log-likelihood (NLL) of the data under this model is given by D d=1 NLL d , where Similar to [10], we infer the parameters (θ, σ) and latent variables X by minimizing NLL using stochastic gradient-based optimization.
Sentence 2: we update the parameters and latent variables with a randomly selected batch of datasets B as where t denotes training iteration, η is the learning rate, N n denotes the total number of observations for pipeline n, and N Bn denotes the number of observations for pipeline n in the batch (we use the convention ∂NLL d ∂xn = 0 when the entry y n,d is unobserved).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The LN model has two important features.
Sentence 2: first, the spike trains of the cell are given by a conditionally (inhomogeneous) Poisson process given x; that is, there are no dynamics in this model beyond those induced by x and K. Second, equation (1) implies: the conditional probability of firing is constant along (hyper)planes in the input space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given a risk parameter δ ∈ (0, 1], we aim for a δ-correct algorithm, that satisfies P µ (µ ∈ H m) ≥ 1 − δ for all µ ∈ H. Our goal is to build δ-correct algorithms that use a small number of samples τ δ in order to reach a decision.
Sentence 2: we want the sample complexity E µ [τ ] to be small.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If the data are i.i.d., then the (B, κ)-Bernstein condition is satisfied with κ = 1 and B = 2λmax µ , where λ max is the maximum eigenvalue of E [xx ⊺ ] and µ = E[yx], provided that µ > 0.
Sentence 2: if x t is uniformly distributed on the sphere and y t = sign(⟨ū, x t ⟩) is the noiseless classification of x t according to the hyper-plane with normal vector ū, then B ≤ c √ d for some absolute constant c > 0.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that as long as p(x) > 0 implies q(x) > 0, any choice of q will yield consistent risk estimates because weighting factors account for the discrepancy between sampling and test distribution (Equation 3).
Sentence 2: ∆n,q is guaranteed to converge to ∆ as n grows large; any approximation employed to compute q * will only affect the number of test examples required to reach a certain level of estimation accuracy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each slice is then whitened [5], and finally we perform sparse k-means on each slice separately, k denotes an integer which is a parameter of our algorithm.
Sentence 2: letting X ∈ R n×p denote a slice of data (n: number of data points and p: dimensionality of the slice) and C n,k denote the set of cluster assignment matrices: Warm-starting k-means with warm started NMF.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Fortunately, we can exploit the following key property of our convex activity shaping framework: the instantaneous average intensity only depends on Ψ(t) through matrix-vector product operations.
Sentence 2: we start by using Theorem 3 to rewrite the multiplication of Ψ(t) and a vector v as We then get a tractable solution by first computing e (A−ωI)t v efficiently, subtracting v from it, and solving a sparse linear system of equations, (A − ωI)x = e (A−ωI)t v − v , efficiently.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This popularity is expressed by the total number of occurrences for the feature k which is given by m k = n−1 i=1 z ik .
Sentence 2: we draw z nk from N B(z nk ; m k , n n+1 ) which has a mean value equal to m k n . Once we have sampled from all represented features we need to consider a sample from the set of unrepresented features.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 3 When they are used for evaluating the risk (e.g., in cross-validation), is by default the zero-one loss, namely 01 (t, y) = (1 − sign(ty))/2; when used for training, 01 is replaced with a surrogate loss [27].
Sentence 2: [15] showed that if satisfies a symmetric condition: we will have which can be minimized by separating X p and X u with ordinary cost-sensitive learning.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This definition is based on the orientation tuning of the input currents to the excitatory model cells in the orientation domain (0.6 < map OSI < 0.9).
Sentence 2: if the sum of input currents is positive (negative) for all presented orientations, recurrent excitation (inhibition) is dominant, and the regime thus excitatory (EXC; respective inhibitory, INH).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Numerical evaluations of bias and variance of estimators discussed in the present section only rely on the computation of coefficients θ α,β defined by where α, β are non-negative integers and (Z, Z ) are bivariate normal (1) with r = ρ.
Sentence 2: we have E ρ [ ρ lin ] = θ 1,1 , Var ρ ( ρ lin ) = (θ 2,2 − θ 2 1,1 )/k.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then M * (x) is a nonempty and compact interval for P X -almost all x ∈ X.
Sentence 2: given a function f : X → R, Lemma 2.3 shows that for P X -almost all x ∈ X there exists a unique t * (x) is the element in M * (x) that has the smallest distance to f (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Following the method described in [11,12], we used AdaBoost during training to select a small set of SIFT features from among all the SIFT features computed for each sample in the training set.
Sentence 2: each training image was represented by a set of SIFT features {F i,j , j = 1, ...n i }, where n i is the number of SIFT features in sample I i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In an nCRP model of personal choice each individual picks a tree path by walking from the root node down the tree, from node to node.
Sentence 2: when situated at a particular parent node, the child node c i individual i chooses is modeled as a random variable that can be either an existing node or a new node: (i) the probability that c i is an existing child node k is proportional to the number of persons who already chose node k from the same parent, (ii) and a new node can be created and chosen with probability proportional to γ 0 > 0, which is the nCRP concentration parameter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Nonetheless, generative classifiers also have several advantages, among them straightforward EM methods for handling missing data, and often better performance when training set sizes are small.
Sentence 2: it has been shown that a simple generative classifier (naive Bayes) outperforms its conditionally-trained, discriminative counterpart (logistic regression) when the amount of available labeled training data is small [11].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We use advantage actor critic (A2C) to train the above reinforcement learning policy.
Sentence 2: let θ = {W i } be the parameters in graph memory representation f (•;θ), and φ be the parameter used in π(a t |T (<t) ,G;φ), our objective is to maximize the expected policy reward: The baseline function b(T (<t) ,G;ψ) parameterized by ψ is used to estimate the expected return, so as to reduce the variance of policy gradient.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we introduce TUCRL, the first algorithm able to perform efficient exploration-exploitation in any finite Markov Decision Process (MDP) without requiring any form of prior knowledge.
Sentence 2: for any MDP with S C communicating states, A actions and Γ C ≤ S C possible communicating next states, we derive a O(D C √ Γ C S C AT ) regret bound, where D C is the diameter (i.e., the length of the longest shortest path between any two states) of the communicating part of the MDP.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If the image is modified by the addition of a distortion vector, x + αû, where û is a unit vector, and scalar α controls the amplitude of distortion, the model can be used to predict the threshold at which the distorted image can be reliably distinguished from the original image.
Sentence 2: one can express a lower bound on the discrimination threshold in direction û for any observer or model that bases its judgments on r (Seriès et al.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We hypothesize that a simple fixed architecture of excitatory and/or inhibitory connections is sufficient to account for the cyclic trends in discrimination thresholds.
Sentence 2: we propose that a recurrent connection profile whose strength varies as a function of (a) the similarity between preferred motion patterns and (b) the distance between receptive field centers, is computationally sufficient to recover the trends in GMP/COM performance (Fig.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Examples of functions that satisfy this assumption will appear later in this section.
Sentence 2: the gradient, as a map from R n (equipped with ≤ K ) to R n (equipped with ≤ K * ), is order-reversing.When ψ is twice differentiable, assumption 1 is equivalent to w, ∇ 2 ψ(u)v ≤ 0, for all u, v, w ∈ K. For example, this is equivalent to Hessian being element-wise non-positive when K = R n + .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assume that the loss function is 1-Lipschitz continuous, then the dependence of the corresponding generalization bound on the architecture for each N k,d k p,q,wid  We present a general framework for capacity control on WN-DNNs.
Sentence 2: we provide a satisfying answer for the central question: we obtain the generalization bounds for L 1,∞ WN-DNNs that grows with depth by a square root term while getting the approximation error controlled.
Relationship: entailment