sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In this section, we present the empirical performance of DECO via extensive numerical experiments.
Sentence 2: we compare DECO after 2 stage fitting (DECO-2) and DECO after 3 stage fitting (DECO-3) with the full data lasso (lasso-full), the full data lasso with ridge refinement (lasso-refine) and lasso with a naive feature partition without decorrelation (lasso-naive).
Relationship: entailment
Domain: NeurIPS
Sentence 1: "Superfast" Numerical Linear Algebra: The structure in these matrices can be exploited for faster linear algebraic operations such as matrix-vector multiplication, inversion and factorization.
Sentence 2: the matrix-vector product can be computed in time O(n log n) for Toeplitz and Hankel matrices, and in time O(n log 2 n) for Vandermonde and Cauchy matrices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As spectral clustering relies on the eigenvectors corresponding to the k smallest eigenvalues (see Algorithm 1) we derive here necessary and sufficient conditions such that in expectation the eigenvectors χ i , i = 1, . . . , k correspond to the k smallest eigenvalues of the normalized Laplacians introduced so far.
Sentence 2: we observe that condition E G affects the ordering of the eigenvalues of the normalized geometric mean Laplacian.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The law of G * t is given by a gamma process, while the masses are conditionally gamma, The idea of [13] is to define the conditional law of G t+1 given G t and C t to coincide with the conditional law of G t given C t as in Proposition 3.
Sentence 2: define where G * t+1 ∼ Γ(α, τ + φ t , H) and w t+1,k ∼ Gamma(c tk , τ + φ t ) are mutually independent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, the total cost could actually be worse than quadratic since the cost of updating each connection could depend on N . Fortunately, the complexity of our collapsed Gibbs sampling algorithm only depends on the number of incident connections, d, or equivalently, the sparsity ρ = d/N .
Sentence 2: we must solve a linear system of size d, which incurs a cubic cost, as seen in Fig.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If for all y ∈ Y, d ∈ D we have p Y,D (y, d) > 0, J p Ŷ |D (y|d), p Y T (y) ≤ , where J is given in (3), and Proposition 2 guarantees that, as long as n is sufficiently large, the utility and discrimination control guarantees will approximately hold when p X, Ŷ |Y,X,D is applied to fresh samples drawn from q D,X,Y .
Sentence 2: the utility and discrimination guarantees will converge to the ones used as parameters in the optimization at a rate that is at least 1 n log n. The distortion control guarantees (4) are a property of the mapping p X, Ŷ |Y,X,D , and do not depend on the distribution of the data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 4 Where µ is a vector of probabilities: This work proposes to instead explicitly parameterize a distribution over distributions on a simplex, p(µ|x ⇤ , ✓), using a DNN referred to as a Prior Network and train it to behave like the implicit distribution in the Bayesian approach.
Sentence 2: when it is confident in its prediction a Prior Network should yield a sharp distribution centered on one of the corners of the simplex (fig.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let > 0, we in turn have This last probability can be bounded by applying McDiarmid's inequality: changing any Y i value changes ψ(Y x,k ) by at most w i,k • t Y (δ 0 ) when we condition on the event Y δ0 .
Sentence 2: this, followed by a union-bound yields Combining with the above we get let δ 0 = δ/2n, with probability at least 1 − δ, for all x ∈ X and k ∈ [n] where the second inequality is an application of Jensen's.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although large entries in a distance matrix A cannot hide, we show it is still impossible to achieve the relative error guarantee in less than mn time for distance matrices.
Sentence 2: we show for the ∞ distance metric, that there are instances of distance matrices A with unequal P and Q for which even for k = 2 and any constant accuracy , must read Ω(mn) entries of A.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The first i,1 mistakes are made at the first step, [ i,2 − i,1 ] + mistakes are made at the second step, + mistakes are made at the third step, and so on.
Sentence 2: the number of mistakes at step k is [ i,k − max k <k i,k ] + and the associated loss is G k . Our main claim follows from combining the upper bounds in Equations 18 and 19.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 6) and exhibit high agreement in independent simulations.
Sentence 2: the goal is to recover the moments of two statistics: the admixture distance, a measure of two individuals' dissimilarity in their ancestry, and the admixture level where 0 means an individual's alleles all come from a single population, and 1 means its ancestry is shared equally among the K populations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each new goal g is drawn in an -greedy fashion (Algorithms 1 & 2) with the exploration probability 2 annealed as learning proceeds (from a starting value of 1).
Sentence 2: in the controller, at every time step, an action is drawn with a goal using the exploration probability 1,g , which depends on the current empirical success rate of reaching g. if the success rate for goal g is > 90%, we set 1,g = 0.1, else 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 3 Known time horizon The two-target algorithm consists in exploring new arms until two successive targets 1 and 2 are reached, in which case the current arm is exploited until the time horizon n. The first target aims at discarding "bad" arms while the second aims at selecting a "good" arm.
Sentence 2: using the names of the variables indicated in the pseudo-code below, if the length L of the first run of the current arm I is less than 1 , this arm is discarded and a new arm is selected; otherwise, arm I is pulled for m − 1 additional runs and exploited until time n if the total length L of the m runs is at least 2 , where m ≥ 2 is a fixed parameter of the algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We derive an approximate predictive posterior for GPPVAE that enables out-of-sample predictions of high-dimensional samples.
Sentence 2: given training samples Y , object feature vectors X, and view feature vectors W , the predictive posterior for image representation y of object p in view q is given by encode training data dz dZ( 12) where x and w are object and feature vectors of object p and view q respectively, and we dropped the dependency on parameters for notational compactness.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this case, the activity rate at stage m is current stage (13) Define x m := λ m−1 (τ m ) − u m−1 − µ (and x 0 = 0 by convention) then the intensity due to events of all previous m stages can be written as τm 0 Ae −ω(t−s) dN (s) = x m e −ω(t−τm) .
Sentence 2: x m is sufficient to encode the information of activity in the past m stages that is relevant to future.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Generalized least squares (GLS) is a method for selecting β when the y i are not necessarily independent and may have different variances.
Sentence 2: if we use a linear model with (possibly correlated) mean-zero noise to model the data, i.e., y = xβ + , where β ∈ R is unknown, is a random vector, E[ ] = 0, and Var( |x) = Ω, then the GLS estimator is the best linear unbiased estimator (BLUE) for β [6]-the linear unbiased estimator with the lowest possible variance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Choosing x t = −v t would incur a loss of − T 2 , while the best fixed action in hindsight, the point 2 T . The player's regret would therefore be Intuitively, the flaw of this naïve strategy is that the hint does not give the player any information about the (d − 1)-dimensional subspace orthogonal to v t . Our solution is to use standard online learning machinery to learn how to act in this orthogonal subspace.
Sentence 2: on round t, we use v t to define the following virtual loss function: In words, we consider the 1-dimensional subspace spanned by v t and its (d − 1)-dimensional orthogonal subspace separately.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since y i ∈ A, we have z i ∈ A M,R/4 , and hence the ball B M (z i , R/4) lies completely inside A M,R/2 ⊆ A M,σ−r .
Sentence 2: the density inside the ball is at least λ everywhere, and hence the mass inside it is at least Observe that R ≥ 4r and so this condition is satisfied as a consequence of satisfying Equation 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The structural predictability model puts us in a position to ask whether empirically observed patterns of syntactic reduction give evidence for speakers' use of some types of cues but not others.
Sentence 2: there is a question of whether predictability based on surface cues alone (the NGRAM features of Section 5.2) provides a complete description of information-density effects on speakers' choices in syntactic reduction.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The variational E-step (see [20] for details) calculates the variational parameters z ij , φ i,j (ρ 1:τ |β 1:τ ) = φ i,j β1 (ρ 1 ) τ t=2 φ i,j βt (ρ t |ρ t−1 ), and η (i,β),(j,ρ) for the lower bounds in (9) (13) (10).
Sentence 2: given the nesting of the lower bounds, we proceed by first maximizing the GMM lower bound L (i,βt),(j,ρt) GM M for each (i, j, β t , ρ t ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We include a factor sp, tp ∈ E if sw 1 − d ≤ tp ≤ sw 2 +d.
Sentence 2: letting π sp be the actual words in sp, the potentials θ sp,tp (x sp , x tp ) determine the preference of the phrase translation π sp , x tp using a phrase table feature function τ : π, π → R k . θ sp,tp (x sp , x tp ) = γ p τ ( π sp , x tp ) if x sp = tp and a large negative value otherwise, where γ p is the weight vector for the Moses phrase table feature vector.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Intuitively, U-Net-1 to U-Net-k all have the same input size, while U-Net-1 views the smallest region with the highest resolution and U-Net-k views the largest region with the lowest resolution.
Sentence 2: for any Second, we need to propagate the higher level information extracted by U-Net-t (2 ≤ t ≤ k) to the next submodule, i.e., U-Net-(t − 1), so that clues from a coarser scale can assist the work in a finer scale.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this work, we study the multi-factor NMF problem based on the generalized KL divergence.
Sentence 2: given an m × n nonnegative matrix V , we seek This simple formulation has a drawback as it is invariant to relative scalings between the factors: for any γ > 0, we have In other words, there exist infinite number of equivalent solutions, which gives rise to an intrinsic ill-posed nature of the mfNMF problem.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To achieve E[f (x a ) − f (x * )] ≤ for the special problem, the IFO complexity of the proposed algorithm is It is interesting to compare Theorem 5 and Corollary 4 with those existing ones for SGD.
Sentence 2: it was shown by Shamir [13] that given in [13].
Relationship: entailment
Domain: NeurIPS
Sentence 1: The second algorithm is much faster and takes O(kn) time, where k is the number of configurations of the hyper-parameters alpha and beta that are searched over.
Sentence 2: for a 784-1000-784 autoencoder model with mini-batches of size 100 and 50,000 training cases on a GTX 580 GPU, learning takes 1.66 seconds per epoch for standard dropout and 1.73 seconds for our second algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: it subtracts from each of these input signals an estimate x obtained by decoding the output spike trains (fig 1A).
Sentence 2: xi = D ij r j , where D = (D ij ) are the decoding weights and r = (r j ) are the filtered spike trains which obey ṙj = −λr j + o j with o j (t) = k δ(t − t k j ) being the spike train of neuron j and t k j are the times of its spikes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Each set Z i is constructed based on S i .
Sentence 2: z i = ∪ j∈[3m] B i,j , where B i,j is a subset of {x i,j , x i,j , y i,j , y i,j } and is constructed as follows: x i,j ∈ B i,j iff j ∈ S i , and x i,j ∈ B i,j iff j ∈ S i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: As with PageRank, QD-PageRank is determined by iterative evaluation of equation 3 from some initial distribution, and is equivalent to the primary eigenvector of the transition matrix Z q , where .
Sentence 2: although P q (i→j) and P q '(j) are arbitrary distributions, we will focus on the case where both probability distributions are derived from R q (j), a measure of relevance of page j to query q: when choosing among multiple out-links from a page, the directed surfer tends to follow those which lead to pages whose content has been deemed relevant to the query (according to R q ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The parameters π are generated by a stick-breaking process where ν i ∼ Beta(α, 1).
Sentence 2: this process results in a decreasing sequence of probabilities π k . given a finite dataset, the probability of seeing feature k decreases exponentially with k. We consider the multi-way classification, where each training data is provided with a categorical label y, where For binary classification and regression, similar procedure can be applied to impose large-margin constraints on posterior distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Example 1 (Linear model with nonlinear Bayes rule): In this scenario, we have two normally distributed classes with different covariance matrices.
Sentence 2: denote X|Y = j ∼ N (µ j , Σ j ) for j = ±1, then µ −1 = (−2, 1) T , µ 1 = (1, 0) T , and .
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, negative equivalence constraints violate this assumption, as dependencies between the hidden variables are introduced.
Sentence 2: assume we have a group Ω = {(a 1 i , a 2 i )} P i=1 of index pairs corresponding to P pairs of points that are negatively constrained, and define the event E Ω = {Y complies with the constraints}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here we show in a rigorous mathematical treatment how homeostatic processes, which have previously received little attention in this context, can overcome common theoretical limitations and facilitate the neural implementation and performance of existing models.
Sentence 2: we show that homeostatic plasticity can be understood as the enforcement of a 'balancing' posterior constraint during probabilistic inference and learning with Expectation Maximization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, identifying which models of neural dynamics can (or cannot) reproduce electrophysiological or imaging measurements of neural activity has been a major challenge [1].
Sentence 2: many models of interest -such as multi-compartment biophysical models [2], networks of spiking neurons [3] or detailed simulations of brain activity [4] -have intractable or computationally expensive likelihoods, and statistical inference has only been possible in selected cases and using model-specific algorithms [5,6,7].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Like for PCA [3], independent components analysis [1], and mixtures of factor analysers [5], the variational Bayesian algorithm for state-space models can be used to learn the structure of the model as well as average over parameters.
Sentence 2: using F it is possible to compare models with different state-space sizes and optimise the dimensionality of the state-space, as we demonstrate in the following section.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [9] exploit group norm sparsity priors to learn dictionaries of natural images by first clustering the training image patches, and then learning a dictionary where members of the same cluster are encouraged to share sparsity patterns.
Sentence 2: they use the 2,1 norm defined as ψ(H) = k ||h k || 2 , where h k are the elements of H that are members of the k-th group.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, it was recently discovered that by restricting attention to a sub-class of tensors such as low-CP rank orthogonal tensors [1] or low-CP rank incoherent 1 tensors [2], one can efficiently find a provably approximate factorization.
Sentence 2: exact recovery of the factorization is possible for a tensor with a low-rank orthogonal CP decomposition [1].
Relationship: entailment
Domain: NeurIPS
Sentence 1: As a result, if max θ∈A λ | θ, xj | < mλ, we can also conclude that [β * λ ] j = 0 by (R1).
Sentence 2: (R1) can be relaxed as In this paper, (R1 ) serves as the foundation for constructing our screening rules, Slores.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let M be a (p, g)-black box for query class F. Then for any database stream X and stream of linear queries F over F, BBSCHEDULER(X, F, M, ✏, , , n, p, g) is (✏, )-differentially private for ✏ < 1 and (↵, )-accurate for sufficiently large constant C and For concreteness, we instantiate this general result with SMALLDB [BLR08], a differentially private algorithm for generating a synthetic database y that closely approximates a true database x on a every query from some fixed set F of k linear queries.
Sentence 2: sMALLDB outputs some y :  In some applications it is more natural for accuracy bounds to improve as the database grows.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2, the two algorithms share similar theoretical guarantees with potential advantages of one on the other depending on the specific setting.
Sentence 2: fLIPFLOP performs slightly better in settings 2, 3, and 4, whereas (A, B)-PROD obtains smaller regret in setting 1, where the constants in the FLIPFLOP bound show their teeth.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This applies to a large variety of complex networks (e.g., social and biological networks) as well as to data sets engineered as networks via similarly graphs, where one often attempts to get a first impression on the data by trying to identify groups with similar behavior.
Sentence 2: finding communities allows one to find like-minded people in social networks, to improve recommendation systems, to segment or classify images, to detect protein complexes, to find genetically related sub-populations, or discover new tumor subclasses.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The Higgins' metric [7] is defined as the accuracy that a low VC-dimension linear classifier can achieve at identifying a fixed ground truth factor.
Sentence 2: for a set of ground truth factors {v k } K k=1 , each training data point is an aggregation over L samples: l |, where random vectors z l are drawn i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, the perspective we are promoting here is that probability distributions defined by graphical models should be viewed as multi-linear functions, and the construction of jointrees should be viewed as a process of constructing arithmetic circuits that compute these functions.
Sentence 2: the fundamental object being factored is a multi-linear function, and the fundamental result of the factorization is an arithmetic circuit.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As in [6], we formulate the problem of learning a fair set of predictions as a convex program.
Sentence 2: our objective is to minimize the expected loss E x,y⇠D [L(f (x), y)], subject to the multifairness constraints defined by C. 5 we show that a simple variant of stochastic gradient descent due to [20] learns such linear families efficiently.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The individual weights are chosen independently and randomly from a Gaussian distribution with mean and variance given by [J ij ] J = 0 and J 2 ij J = g 2 /N , where square brackets are ensemble averages [9 -11,13].
Sentence 2: the control parameter g which scales as the variance of the synaptic weights, is particularly important in determining whether or not the network produces spontaneous activity with non-trivial dynamics (g = 0 corresponds to a completely uncoupled network and a network with g = 1 generates non-trivial spontaneous activity [4,9,10]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Define See Figure 2  We conjecture that the difference Right γ n,k − Left γ n,k can be used as before to construct a density estimate.
Sentence 2: if γ n,k is a shortest path from the anchor point X 0 to X s , we believe that under similar conditions on k and n as before, is a consistent estimator of the quantity log p(X s ) − log p(X 0 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Lemma 3 The function f Θ : R M → R M defined above is permutation equivariant iff all the offdiagonal elements of Θ are tied together and all the diagonal elements are equal as well.
Sentence 2: this result can be easily extended to higher dimensions, i.e., X = R d when λ, γ can be matrices.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We show that this approach provides a natural trajectory, in which simple, cheap classifiers are applied initially, and used to provide guidance on which of our more expensive classifiers is likely to be more informative.
Sentence 2: we show that we can get comparable (or even better) performance to a method that uses a large range of expensive classifiers, at a fraction of the computational cost.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Not all probability distributions can be efficiently sampled.
Sentence 2: high-dimensional distributions such as the uniform distribution on the cube [0, 1] d are especially susceptible to sampling difficulties due to the curse of dimensionality, where in some cases it is necessary to take exponentially many (in the dimension of the space) sample points in order to obtain a satisfactory estimate.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We apply this convex duality framework to the f-divergence and Wasserstein distance families, providing interpretation for f-GAN, including vanilla GAN minimizing the JS-divergence, and Wasserstein GAN.
Sentence 2: we generalize [1]'s interpretation of the vanilla GAN problem (1), which only holds for an unconstrained discriminator set, to the more general case with linear space discriminator sets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assume that we have a ground-truth distribution over the subspace spanned by the first d dimension of the Euclidean space R D . We will assume that this distribution is a single Gaussian N (0, λ 2 I d ).
Sentence 2: suppose this distribution is corrupted with Gaussian noise along the orthogonal subspace of dimension D − d. for any "true" point x drawn from N (0, λ 2 I d ), its observation x e is drawn from N (x, σ 2 I D−d ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We have shown that Exploration and Hybrid 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 of Hybrid can be boosted to near the minimax optimal sample complexity of full information algorithms.
Relationship: entailment