sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Unlike the previous example, our analysis shows that here the reconstruction of both the prior and the characteristics of sensory noise is relatively reliable (Figure 3 and Supplementary Material), without major biases, even when we decouple the internal representation of the noise from its objective counterpart (except for underestimation of the noise lower bound s 0 , and of the internal noise σHigh , Figure 3 c).
Sentence 2: in all cases the exponent k prior of the prior over speeds can be recovered with good accuracy.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the reweighted Bethe setting, we also present a useful characterization of the concave region of the Bethe entropy function in terms of the geometry of the graph.
Sentence 2: we show that if the region graph consists of only singleton vertices and pairwise edges, then the region of concavity coincides with the convex hull of incidence vectors of single-cycle forest subgraphs of the original graph.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is not hard to see that the equivalence relation over complete parent configurations, K = ΦΦ , is equal to the componentwise (Hadamard) product of the individual equivalence relations for each parent variable.
Sentence 2: unfortunately, the dual problem ( 4) is still expressed in terms of the indicator matrix Y over child variable values, which is still not acceptable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: At each time point t we may have a corpus of documents {y nt t } Nt nt=1 , where y nt t is a P -dimensional observation vector, and N t denotes the number of documents at time t. The model presented in Section 2.1 is readily extended to handle this situation.
Sentence 2: at each time point t, for each document n t , the ERL representation for word count p, denoted by y nt p,t , is y nt p,t = g z nt p,t , p = 1, . . . , P, t = 1, . . . , T, n t = 1, . . . , N t , where z nt t ∈ R P and P is the vocabulary size.
Relationship: entailment
Domain: NeurIPS
Sentence 1: measures optimized by traditional compression algorithms, such as peak signal-to-noise ratio (PSNR), they avoid artifacts such as blur and blocking, producing visually more pleasing results [1][2][3][4][5][6][7][8][9][10].
Sentence 2: visual quality can be improved by incorporating generative adversarial networks (GANs) [14] into the learning process [4,15].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our result is an easily computable distribution p which comes within a factor of 3 of " 1 p and, as a result, within a factor of 9 in terms of sample complexity (in practice we expect this to be even smaller, as the factor of 3 comes from consolidating bounds for a number of different worst-case matrices).
Sentence 2: to put this in perspective note that the definition of p does not place any restriction either on the access model for A while computing p , or on the amount of time needed to compute p . we are competing against an oracle which in order to determine p has all of A in its purview at once and can spend an unbounded amount of computation to determine it.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2 Given a parametric model M, a maximum likelihood estimator (MLE) is a function f MLE : S → Θ such that for any data P ∈ S, f MLE (P ) is a parameter that maximizes the likelihood of the data.
Sentence 2: f MLE (P ) ∈ arg max θ∈Θ Pr(P |θ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To simplify the notation we write k T x f for f, k x H , and f T Lg for f, Lg , where f, g ∈ H and L : H → H, even if H is infinite-dimensional.
Sentence 2: a Gaussian process GP(m, k) has a dual representation in an RKHS H [12]: there exists µ ∈ H and a positive semi-definite linear operator Σ : H → H such that for any x, x ∈ X , ∃φ x , φ x ∈ H, the mean function has a realization µ in H, which is defined by the reproducing kernel k(x, x ) = ρ 2 φ T x φ x ; the covariance function can be equivalently represented by a linear operator Σ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this case, when F G contains all Borel measurable distributions on X , minimax bounds have been established under very general conditions (essentially, when the sample space X is an arbitrary totally bounded metric space) in terms of covering numbers of X [50,39,23].
Sentence 2: in the particular case that X is a bounded subset of R d of full dimension (i.e., having non-empty interior, comparable to the case X = [0, 1] d that we study here), these results imply a minimax rate of Sobolev IPMs: The closest work to the present is [26], which we believe was the first work to analyze how convergence rates jointly depend on (Sobolev) smoothness restrictions on both F D and F G . for Sobolev spaces F D = W s,p and F G = W t,q with p, q ≥ 2 (compare our Example 4), they showed Our main results in Sections 4 and 5 improve on this in two main ways.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The solution ŵ can therefore be interpreted as the superposition of a vector which has small `2 norm, and a vector which has small k-support norm, with the parameter a regulating these two components.
Sentence 2: as a tends to zero, in order to prevent the objective from blowing up, û must also tend to zero and we recover k-support norm regularization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For example, a sub-tree 'a-b-c-d', where 'a', 'b', 'c' and 'd' represent symbols and '-' represents an edge between symbols, covers sub-trees 'd', 'a-c-d' and 'b-d'.
Sentence 2: c u (t) = λ if u matches t allowing the node skip, 0 otherwise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This concept was introduced by InfoGAN [14], where latent codes are utilized to discover the salient semantic features of the data distribution and guide the generating process.
Sentence 2: the regularization seeks to maximize the mutual information between latent codes and trajectories, denoted as I(c; ⌧ ),which is hard to maximize directly as it requires access to the posterior P (c|⌧ ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Crucially, we show that learning an optimal Bayesian network structure with vertex cover number at most k can be done in polynomial time for any fixed k. Moreover, vertex cover number provides an upper bound for tree-width, implying that inference is also tractable; thus, we identify a rare example of a class of Bayesian networks where both learning and inference are tractable.
Sentence 2: our main theoretical result shows that an optimal Bayesian network structure with vertex cover number at most k can be found in time 4 k n 2k+O(1) (Theorem 5).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The methods above are typically applied to imaging data offline, after the experiment is complete; however, there is a need for accurate and fast real-time processing to enable closed-loop experiments, a powerful strategy for causal investigation of neural circuitry [10].
Sentence 2: observing and feeding back the effects of circuit interventions on physiologically relevant timescales will be valuable for directly testing whether inferred models of dynamics, connectivity, and causation are accurate in vivo, and recent experimental advances [11,12] are now enabling work in this direction.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Informally, the doctrine codifies the notion that not only should protected attributes, such as race and gender, not be explicitly used in making decisions, but even after the decisions are made they should not be disproportionately different for applicants in different protected classes.
Sentence 2: if an unprotected feature, for example, height, is closely correlated with a protected feature, such as gender, then decisions made based on height may still be unfair, as they can be used to effectively discriminate based on gender.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is accomplished by employing a grid search over λ and a combined grid/iterative search over σ 2 .
Sentence 2: for each fixed grid value of λ we seek a minimizer over σ 2 by evaluating the validation risk at a coarse grid of σ 2 values and then performing a Golden search over the interval defined by the two σ 2 values on either side of the coarse grid minimum.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The empirical estimation of R(f ) (also p f (0)) can be obtained by kernel density estimation, which is defined as: Hence, the approximation of f * can be found by learning algorithms associated with R σ z (f ).
Sentence 2: in theory, for any f : there holds R(f In this paper, we assume that f * (x) = mode(Y |X = x) = w T * x for some w * ∈ R p . Following the ideas of ridge regression and Lasso [22], we consider the robust linear estimator for learning the conditional mode function.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We start with a low-dimensional mapping of the original data that is guaranteed to preserve the value of a shift-invariant kernel (specifically, the random Fourier features of Rahimi and Recht [8]), and convert this mapping to a binary one with similar guarantees.
Sentence 2: we show that the normalized Hamming distance (i.e., Ham-ming distance divided by the number of bits in the code) between any two embedded points sharply concentrates around a well-defined continuous function of the kernel value.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Any f ∈ L 2 (Ω) which can be -approximated with polynomial in 1/ number of steps of gradient descent is infinitely differentiable.
Sentence 2: f function must belong to the intersection of all Sobolev spaces on Ω. Gradient descent for periodic functions on R. Let us now consider a simple but important special case, where the reach can be analyzed very explicitly.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To model the responses of thousands of neurons in a free viewing experiment, we take an alternative approach and directly estimate a receptive field shift for all neurons from the tracked pupil position solely based on optimizing the predictive performance of the network.
Sentence 2: we feed the pupil location p ∈ R 2 into a network that predicts a shift ∆x = (∆x, ∆y) ∈ R 2 for each time point which is added onto all locations x i = (x i , y i ) of the spatial transformer readout.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is not hard to see that the set {� pair σ : σ ∈ T } spans a r(r−1) 2 dimensional space, and hence the nullity of the above matrix is at most r(r−1)− � r(r−1) Thus by Theorem 13, we get that CCdim(� pair ) ≥ r(r − 1) − � r(r−1) 2 − 2 .
Sentence 2: for r ≥ 5, this gives CCdim(� pair ) > r, and therefore establishes that no convex surrogate ψ operating on a surrogate target space � T ⊆ R r can be classification calibrated with respect to � pair on the full probability simplex Δ r(r−1) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, a significant amount of research has been carried out on sparse PCA recently, and we thus propose to leverage some of the recently proposed relaxations and heuristics to solve this rank 1 sparse PCA problem (see [8,20] and references therein).
Sentence 2: the Truncated Power iteration (TPI) algorithm of [20] can easily be modified to compute Ω * k,q which corresponds to a generalization of the rank 1 sparse PCA in which in general a = b and k = q.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this way, we develop a set of grid points that is typically a very small subset of the entire possible grid, which is adapted to the parts of the belief space typically inhabited by the agent.
Sentence 2: given a grid resolution r and a belief state b we can compute the coordinates (grid points g i ) of the belief simplex that contains b using an efficient method called Freudenthal triangulation [2].
Relationship: entailment
Domain: NeurIPS
Sentence 1: As α, F p , F n and b are all constants for the purpose of this analysis, we may optimize for b 1 in the equivalent expression The derivative with respect to b 1 is This equals 0 when This further yields that the false positive rate is minimized when b 2 = b * 2 , where This result may be somewhat surprising, as here we see that the optimal value b * 2 is a constant, independent of b.
Sentence 2: the number of bits used for the backup filter is not a constant fraction of the total budgeted number of bits bm, but a fixed number of bits; if the number of budgeted bits increases, one should simply increase the size of the initial Bloom filter as long as the backup filter is appropriately sized.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A fragmentation-coagulation process (FCP) is a continuous-time Markov process π ≡ (π(t), t ∈ [0, T ]) over a time interval [0, T ] where each π(t) is a random partition in Π S . Since the space of partitions for a finite S is finite, the FCP is a Markov jump process (MJP) [18] : it evolves according to a discrete series of random events (or jumps) at which it changes state and at all other times the state remains unchanged.
Sentence 2: the jump events in an FCP are either fragmentations or coagulations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The unnormalized relative entropy projection step can be performed efficiently by using a procedure similar to the Sinkhorn balancing used for the Birkhoff polytope in [12].
Sentence 2: given a nonnegative matrix X ∈ R m×n + , one alternately scales the rows and columns to match the desired row and column sums until some convergence criterion is met.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The proportion of the mass that a parent j at level k − 1 transmits to a given child a depends on the weight w ja between the parent and the child (normalized by the sum of the weights of the edges from j to all its children), and the probability mass P k−1 SS (j) that is assigned to node j.
Sentence 2: the portion a child event a is able to obtain from the total discounted mass depends on how authoritative its parents are, and how strong the relationship between the child and its parents.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the last stage, we obtain the generated high-resolution image after passing the last hidden state h m−1 through a convolutional layer with 3 × 3 kernel size.
Sentence 2: the drawing bot, denoted as Î = C(S, z), is decomposed as h 0 = F 0 (z, F ca (e)); Here, z is a noise vector usually sampled from a standard normal distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Based on the correctness of this procedure, we describe a dynamic program to compute the choice probabilities of a general offer set S. The key idea is to decompose these probabilities to include the position at which a product is chosen.
Sentence 2: for i  k and s 2 [k], let ⇡(i, s, k) be the probability that product a i is chosen (i.e., first among products in S) at position s after the k-th step of the RIM.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Similar to Lemma 1, by making additional assumptions concerning the prediction method used we can derive a slightly simpler bound.
Sentence 2: for a labeled set L that is an α cover, we assume unlabeled nodes are labeled with the weighted majority vote of neighbors in L. In other words, set ŷi = y i for i ∈ L, and set ŷi = y for i / ∈ L with y such that j∈L:y j =y W i,j ≥ j∈L:y j =y W i,j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Similar to deep network approaches [11,14], recent works tried to avoid feature engineering and employed sparse coding of image patches to learn features from "scratch", for texture analysis [15] and object recognition [30,3].
Sentence 2: orthogonal Matching Pursuit [18] is a greedy algorithm that incrementally finds sparse codes, and K-SVD is also efficient and popular for dictionary learning.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In contrast, in this paper we propose a novel generative model that is able to reason jointly about the 3D scene layout as well as the 3D location and orientation of objects in the scene.
Sentence 2: given a video sequence of short duration acquired with a single camera mounted on a moving car, we estimate the scene topology and geometry, as well as the traffic activities and 3D objects present in the scene (see Fig.
Relationship: entailment
Domain: NeurIPS
Sentence 1: According to this analysis, by setting T = Θ(1/λ), then as long as the total number of batches is at least Ω(log(1/ )), and λ = Ω(1/m), then the algorithm will attain an -suboptimal solution in expectation.
Sentence 2: without any additional communication, we extend the applicability of distributed SVRG (at least for regularized least squares) from the λ = Ω(1/ √ m) regime to 1/λ = Ω(1/m).
Relationship: entailment
Domain: NeurIPS
Sentence 1: An important point to note is that the complexity of these algorithms depends simply on the size k and the number of nodes n. There is no loss in generality in assuming that the size-k FVS F is fully connected and each of the feedback nodes has edges to every non-feedback node.
Sentence 2: after re-ordering the nodes so that the elements of F are the first k nodes (T = V \F is the set of nonfeedback nodes of size n − k), we have that J = J F J T M J M J T 0, where J T 0 corresponds to a tree-structured subgraph among the non-feedback nodes, J F 0 corresponds to a complete graph among the feedback nodes, and all entries of J M may be non-zero as long as ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although the improvement is on the non-dominating term, multi-stage ORDA is an optimal algorithm for both stochastic and deterministic optimization.
Sentence 2: for deterministic strongly convex and smooth f (x) (M = 0), one can use the same algorithm but only replaces the stochastic subgradient G(x, ξ) by the deterministic gradient ∇f (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Generally, FOMs use the DGF to choose steps: given a gradient and a scalar stepsize, a FOM moves in the negative gradient direction by finding the point that minimizes the sum of the gradient and of the DGF evaluated at the new point.
Sentence 2: the next step can be found by solving a regularized optimization problem, where long gradient steps are discouraged by the DGF.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our result shows that, with a suitable choice of the regularization parameter λ, the rescaled estimation error is of the order r max{d 1 , d 2 }/n + √ rd 1 d 2 /n up to a logarithmic factor.
Sentence 2: for n min{d 1 , d 2 }, the leading term of the error is r max{d 1 , d 2 }/n.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inductively suppose we have solved the first tasks through programs stored below address 7 ¦ A ¢ , and that the most recently found program starting at address 7 ¡ £¢ ¥¤ i a7 9 ¦ A ¢ actually solves all of them, possibly using information conveyed by earlier programs.
Sentence 2: to find a program solving the first 8 ¦ tasks, OOPS invokes Try as follows (using set notation for ring W ): we spend roughly equal time on two simultaneous searches.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we consider an interesting and not uncommon scenario of multitask learning, where the tasks are heterogeneous and bear a union support.
Sentence 2: each task can be either a regression or classification problem, with the inputs lying in a very high-dimensional feature space, but only a small number of the input variables (i.e., predictors) are relevant to each of the output variables (i.e., responses).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We choose the coefficients a, b ≥ 0 such that s 0 a 2 + b 2 = 1, whence E{X 2 i,p } = 1 and hence the p-th covariate is normalized as the first (p − 1) ones.
Sentence 2: the rows of X are i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, we show that in the stochastic setting SODA achieves an O a:∆a>0 Kε 2 ∆a pseudo-regret bound that holds simultaneously with the adversarial regret guarantee.
Sentence 2: sODA is safe against an unrestricted oblivious adversary and provides improved regret guarantees for at least two different types of "easiness" simultaneously.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For every d > 0, m 2 N, distribution D on Z, The dependence on the data distribution is captured through t, which is ideally chosen as small as possible, subject to Q(S) yielding small empirical risk.
Sentence 2: (One can use a union bound to tune t based on S.) The fact that the KL bound does not depend on D, other than through t, implies that the bound must be loose for all t such that there exists a distribution D that causes Q to overfit with high probability on size m datasets S ⇠ D m . for fixed t, the bound is no longer distribution dependent.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To further drive intuition behind the E-step complication, let us make the sum-to-one constraint for the variational distributions q n (•) explicit.
Sentence 2: k q n (k) = 1 for all data points n, which according to the above block constraint and using Property 1 can be reformulated as the |L| constraints Notice that since q B k can be associated with an internal node in T it may be the case that q B k(l) represent the same q B k across different constraints in (5).
Relationship: entailment
Domain: NeurIPS
Sentence 1: 4 Fast Stagewise Unit-Rank Tensor Factorization (SURF) For simplicity, we drop the index r and write the generic form of the problem in (6) as ) , where σ ≥ 0, w (n) 1 = 1, and the factors Before diving into the stagewise/boosting algorithm, we first consider an alternating convex search (ACS) approach [Chen et al., 2012, Minasian et al., 2014 which appears to be natural for solving (7) with any fixed tuning parameter.
Sentence 2: we alternately optimize with respect to a block of variables (σ, w (n) ) with others fixed.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our rate-distortion analysis of the Poisson channel differs from that comparison in a twofold way: First, we do not change-the noise model and second, the MMSE is often more appropriate to account for the coding efficiency than the channel capacity [10].
Sentence 2: the assumption of a real random variable to be encoded with minimal mean squared error loss appears to introduce a bias for analog coding rather than for binary coding.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Since the objective only depends on the distribution of the maximum coordinate of ĉt , we can continue the upper bound with a maximum over any distribution of random vectors whose coordinates are 0 with probability at least 1 − K/L and otherwise are −L or L with equal probability.
Sentence 2: let t be a Rademacher random vector, we continue with Now observe that if we denote with a = Pr[Z t = L], the above is equal to We now argue that this maximum is achieved by setting a = K/L.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead, we suggest using an analogue of the augmented R-W model, which uses the above noisy-ORlAND-NOT prediction instead of the linear prediction implicit in the augmented R-W model.
Sentence 2: we define the following algorithm (with all parameters as defined before), using the notational device that the C/ s are preventive and the C j ' s are generative: Unlike the R-W and augmented R-W models, there is no known characterization of the long-run behavior of this iterative algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It is intriguing (A) to see this quantity appear as analogous to the score function in the ICA likelihood model, and, (B) to speculate that experiments could show that this' voltage slope at threshold' is a hidden factor in STDP data, explaining some of the scatter in Figure 3A.
Sentence 2: an STDP datapoint should lie on a 2-surface in a 3D space of {∆w, ∆t, uk }.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 1 For R 3way c-NN one can construct a data structure with O(n ρ log 1/cR0 n) query time and O(n 1+ρ ) space, where ρ = 1 The argument for 2-way resemblance can be naturally extended to k-way resemblance.
Sentence 2: given three sets S 1 , S 2 , S 3 ⊆ Ω and an independent random permutation π : Ω → Ω, we have: Eq.
Relationship: entailment