sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Then the θ-th power of the kernel k is a function k θ : X × X → R defined by This is again a reproducing kernel [22,Proposition 4.2], and defines an RKHS called the θ-th power of the RKHS H: This is an intermediate space between L 2 (Q) and H, and the constant 0 Indeed, H θ is nesting w.r.t.
Sentence 2: θ: H θ gets larger as θ decreases.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This gives the following result: where and It is possible to compute the gradient of log Z with respect to θ kj , i.e., the j-th hyper-parameter of the k-th covariance function used to compute K k . Such gradient is useful to find the covariance functions c k (•, •), with k = 1, . . . , l, that maximize the model evidence.
Sentence 2: one can show that, if EP has converged, the gradient of the free parameters of the approximate factors with respect to θ kj is zero [18].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Integrating out φ and π results in the following expression for sampling z ij : where n (s,i,j,v) = 1(z ij = s, x ij = v).
Sentence 2: if x ij takes feature value v for feature j and is assigned to cluster s, then n (s,i,j,v) = 1, or 0 otherwise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 1 shows that the cross-polytope LSH achieves essentially the same bounds on the collision probabilities as the (theoretically) optimal LSH for the sphere from [2] (see Section "Spherical LSH" there).
Sentence 2: substituting the bounds from Theorem 1 for the cross-polytope LSH into the standard reduction from Near Neighbor Search to LSH [6], we obtain the following data structure with sub-quadratic space and sublinear query time for Near Neighbor Search on a sphere.
Relationship: entailment
Domain: NeurIPS
Sentence 1: for (r, k) ∈ {1, . . . , d} × {1, . . . , K} do The (r, k)-element of f is updated as  To learn the nonlinear tensor factorization model ( 1), we propose to optimize the regularized empirical risk in an alternating way.
Sentence 2: we optimize each component f (r,k) with the other fixed components {f (r ,k ) } (r ,k ) =(r,k) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , p n be random variables such that {p i } i2H0 are independent and sub-uniformly distributed so that max i2H0 P(p i  x)  x.
Sentence 2: for any k 2 {0, 1, . . . , n}, let ⌘  4 log(9/ ) any procedure that chooses a set {i :  by the standard BH result.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Define the quantity where 1) ).
Sentence 2: if we choose the step sizes as γ ≤ 1/σ α and β t as in (15), then (i) the above quantity is upper bounded as C(t) ≤ C for all t ≥ 1, where such that M is an upper bound to M (t) , and (ii) the MCGD method converges to an -optimal solution to (5) in T iterations, i.e., F 0 (α (T ) , Θ (T ) ) − F 0 ( α, Θ) ≤ , where as C(T ) ≤ C, at most C( −1 − (F 0 (α (0) , Θ (0) ) − F 0 ( α, Θ)) −1 ) + iterations are required for the MCGD method to reach an -optimal solution to (5).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Its probability mass function is given by 4 Variational Approximation of Posterior Generally, there are two approaches to learning a mixture model from observed data, namely Maximum likelihood estimation (MLE) and Bayesian learning.
Sentence 2: maximum likelihood estimation seeks an optimal point estimate of ν, while Bayesian learning aims to derive the posterior distribution over the mixtures.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Results for a MCMC-based DPFM of size 64-32, with the same setting used for the first experiment, indicate that pharmaceutical topics derived from this analysis form clinically reasonable clusters of pharmaceuticals, that may be prescribed to patients for various ailments.
Sentence 2: we found that layer-1 topic 46 includes a cluster of insulin products: Insulin Glargine, Insulin Lispro, Insulin Aspart, NPH Insulin and Regular Insulin.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Here one wants to learn a ranking model f : X → R that assigns higher scores to positive instances than to negative ones.
Sentence 2: the goal is to learn a ranking function f with small bipartite ranking error: , where (x, y), (x , y ) are assumed to be drawn iid from D; this is the probability that a randomly drawn pair of instances with different labels is mis-ranked by f , with ties broken uniformly at random.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Inspired by the work in [8] and [9], we propose a semi-supervised learning algorithm for DRFs that makes full use of the available data by exploiting a form of entropy regularization as a prior over the parameters on D u .
Sentence 2: for a semi-supervised DRF, we attempt to find θ that maximizes the following objective function The first term of ( 4) is the conditional likelihood over the labeled data set D l , and the second term is a conditional entropy prior over the unlabeled data set D u , weighted by a tradeoff parameter γ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our starting point is a simple and schematic model of vision problems embodying the idea that images are generated by a set of hidden variables with local dependencies.
Sentence 2: we assume that each observed image element x i is connected to a hidden variable z i by a directed edge, and each hidden variable is connected to its neighbors (in set c i ) by an undirected edge (thus implying that each hidden variable is conditionally independent of all others given its neighbors).
Relationship: entailment
Domain: NeurIPS
Sentence 1: After performing M runs of AIS, the importance weights w AIS can be used to obtain an unbiased estimate of our model's partition function Z S : where Z 0 = 2 F . Observe that the Markov transition operators do not necessarily need to be ergodic.
Sentence 2: if we were to choose dumb transition operators that do nothing, for all s, we simply recover the simple importance sampling procedure of Eq.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For convenience, we define which is a full-dimensional update version of Step 3.
Sentence 2: one can observe that z (k+1) is updated as Notice that from ( 5), ( 6), ( 7) and ( 8) we have which together with (10) yields in Step 4, we only need to update the block coordinates x and set the rest to be y i .
Relationship: entailment
Domain: NeurIPS
Sentence 1: In a key contribution of this paper, we impose a very natural assumption on general listwise ranking functions, which we term exchangeability, which formalizes the notion that the ranking function depends only on the object features, and not the order in which the documents are presented.
Sentence 2: as detailed further in Section 3, we define exchangeable ranking functions as those listwise functions where if their set of input objects is permuted, their output permutation/score vector is permuted in the same way.
Relationship: entailment
Domain: NeurIPS
Sentence 1: ( 1) is equal to λ max (A * k ), where A * k is the k × k principal submatrix of A with the largest maximal eigenvalue.
Sentence 2: the non-zero elements of the optimal sparse factor x are exactly equal to the elements of u * k , the principal eigenvector of A * k . This underscores the inherent combinatorial nature of sparse PCA and the equivalent class of cardinality-constrained optimization problems.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Also Theorems 1 and 2 in [5] show that the logarithmic dependence on |S m | is optimal asymptotically.
Sentence 2: [5, Theorem 1] matches with the growing diameter result in Remark 1(ii), while [5,Theorem 2] shows that if Λ is absolutely continuous w.r.t.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It has been shown that the responses from a population of phase-tuned disparity energy are more comparable than the responses from a population of position-tuned disparity energy neurons [12].
Sentence 2: the preferred disparity of the neuron with maximum response in a population of phase tuned neurons is a more reliable indicator of the stimulus disparity than the preferred disparity of the neuron with maximum response in a population of position tuned neurons, especially for neurons with small phase shifts.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While of interest in its own right, it is also possible to express important performance criteria as particular cases of d α,p .
Sentence 2: the so-called reconstruction error [13]: Note that d R is a natural criterion because a k-truncated PCA estimate minimizes a suitable error d R over all subspaces of dimension k. Clearly, d R (S ρ , Ŝ) vanishes whenever Ŝ contains S ρ and, because the family { Ŝk n } n k=1 of PCA estimates is nested, then d R (S ρ , Ŝk n ) is non-increasing with k. As shown in [13], a number of unsupervised learning algorithms, including (kernel) PCA, k-means, k-flats, sparse coding, and non-negative matrix factorization, can be written as a minimization of d R over an algorithm-specific class of sets (e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Further, the noise in the observations varies over time.
Sentence 2: sometimes the variance of the first observation is Σ v,11 = Var(v t,1 ) = 2 6 The parameters A ∈ R 4×4 and B ∈ R 4×2 are obtained from a standard 747 ("yaw damper") model, which may be found in, e.g., the Matlab control toolbox, and various texts such as [6]. )
Relationship: entailment
Domain: NeurIPS
Sentence 1: Formally, fix s 2 (0, 1) such that 1/s is an integer.
Sentence 2: define M i (s) to be the sub-interval (5.1) the set M s := {M s (i) : i = 1, . . . , 1/s} is a partition of [0, 1) into 1/s sub-intervals of length s each.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Well-known algorithms include MUA [6], projected gradientd descent [7], alternating nonnegative least squares (ANLS) [8], and hierarchical ALS (HALS) [9].
Sentence 2: aNLS (which uses the block principal pivoting algorithm to very efficiently solve the nonnegative least squares) and HALS achive the state-of-the-art performance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For this, we apply another sequence of relational embedding modules.
Sentence 2: the output of the previous object-relational embedding module O 4 ∈ R N×C obj , and the intermediate feature O 3 ∈ R N×256 are taken as inputs as: where the argmax is conducted row-wise and produces an one-hot encoded vector O 4 which is then mapped into R N ×200 by a parameter matrix K 1 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: MOCHA is also equipped to handle heterogeneity from changing systems environments, such as battery power, memory, or network connection, as we show in Figure 2.
Sentence 2: we simulate systems heterogeneity by randomly choosing the number of local iterations for MOCHA or the mini-batch size for mini-batch methods, between 10% and 100% of the minimum number of local data points for high variability environments, to between 90% and 100% for low variability (see Appendix E for full details).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In both cases, the compatibility function becomes Comparing the optimization problems (10)/( 11) and ( 12)/(13) to the ordinary SVM dual, we see that MLSP couples |Y| binary SVM problems by the joint kernel function and the summed-over box constraints.
Sentence 2: whenever only a feasibly small subset of variables has to be considered, we can solve the problem using a general purpose QP solver, or a slightly modified SVM solver.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The probabilities of each left/right transitions in the tree are modeled by a set of binary logistic regressors and the probability of a given word is then obtained by multiplying the probabilities of each left/right choice of the associated tree path.
Sentence 2: let l(v i ) be the sequence of tree nodes on the path from the root to the word v i and let π(v i ) be the sequence of binary left/right choices for each of those nodes (e.g.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If one uses as initialization of RatioDCA, f 0 = 1 A , then either RatioDCA terminates after one step or it yields an f 1 which after optimal thresholding as in Theorem 3.1 gives a partition (B, B) which satisfies The above "improvement theorem" implies that we can use the result of any other graph partitioning method as initialization.
Sentence 2: we can always improve the result of spectral clustering.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A separate computational line of reasoning leads to the claim that some transformations of visual inputs that preserve viewed object identity are class-specific.
Sentence 2: the 2D images evoked by a face undergoing a 3D rotation are not produced by the same image transformation (2D) that would produce the images evoked by an object of another class undergoing the same 3D rotation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: A convex body K is said to be µ-well-conditioned for µ ≥ 1, if there exists a function F : R d → R such that K = {x|F (x) ≤ 0} and for every x ∈ ∂K: This notion of well-conditioning of a convex body to the best of our knowledge has not been defined before, but it intuitively captures the notion that the convex body should "curve" in all directions to a certain extent.
Sentence 2: the unit ball has µ = 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the limiting case where λ tends to 1 and ∞, this cost function reduces to the average depth and worst case depth, respectively.
Sentence 2: as mentioned in Section 2, GBS is tailored to minimize L 1 (Π), and hence may not produce a good suboptimal solution for the exponential cost function with λ > 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Uniform stability of a learning algorithm is a classical notion of algorithmic stability introduced to derive high-probability bounds on the generalization error (Bousquet and Elisseeff, 2002).
Sentence 2: for a loss function with range bounded in [0, 1], the generalization error of a γ-uniformly stable learning algorithm on n samples is known to be within O((γ + 1/n) n log(1/δ)) of the empirical error with probability at least 1 − δ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In essence, if τ is too small, there is insufficient regularization.
Sentence 2: if the minimum expected degree δ = O(ln N ), then we need τ ≥ c( ) ln N to have enough regularization to satisfy condition (b) on δ + τ . Alternatively, if τ is too large, it washes out significant eigenvalues.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , T , which are all 1-Lipschitz with respect to ∆, such that the regret (without movement costs) is Ω T d+1 d+2 .
Sentence 2: in many natural metric spaces in which the upper and lower Minkowski dimensions coincide (e.g., normed spaces), the bound of Theorem 9 is tight up to logarithmic factors in T. and quite surprisingly, we see that the movement costs do not add to the regret of the problem!
Relationship: entailment
Domain: NeurIPS
Sentence 1: The intersection graph, denoted by G int , is a graph whose vertex set is V = {x 1 . . . x n } and x i is adjacent to x j if the sets A i and A j intersect.
Sentence 2: there exists an output of the blackbox that depends both on x i and x j (see Figure 2 for an illustration).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section we establish an explicit tradeoff between the performance of a multi-player algorithm and the number of communication rounds it uses, in terms of the accuracy ε.
Sentence 2: our observation is that by allowing O(log(1/ε)) rounds of communication, it is possible to achieve the optimal speedup of factor k. we do not gain any improvement in learning performance by allowing more than O(log(1/ε)) rounds.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To estimate ARI in this experiment, we used simple least-squares linear regression.
Sentence 2: for each k ∈ {2, . . . , 9}, we fit ARI as a linear function of Silhouette scores using all the data from the meta-training set in the partition pertaining to k: each dataset in the meta-training set provided 10 target values, corresponding to different runs where number of clusters was fixed to k. As is standard, we define the best-fit k * i for dataset i to be the one that yielded maximum ARI score across the different runs, which is often different from k i , the number of clusters in the ground truth (i.e., the number of class labels).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our answer-generator module assumes the answer distribution is independent from the history h t−1 = (a 1:t−1 , q 1:t−1 ).
Sentence 2: we approximate the likelihood as p(a t |c, q t , a 1:t−1 , q 1:t−1 ) ∝ p (a t |c, q t ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To conquer these limitations, we reexamine the original L-BFGS algorithm and propose a new L-BFGS update procedure, called Vector-free L-BFGS (VL-BFGS), which is specifically devised for distributed learning with huge number of variables.
Sentence 2: we replace the original L-BFGS update procedure depending on vector operations, as known as two-loop recursion, by a new procedure only relying on scalar operations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This grid partitions the real number line into intervals [u + nδ, u + (n + 1)δ] for all integers n. The probability that two points x and y fall in the same bin in this grid is max 0, 1 − |x−y| δ [13].
Sentence 2: if we number the bins of the grid so that a point x falls in bin x = x−u δ and y falls in bin ŷ = y−u δ , then Pr u [x = ŷ|δ] = k hat (x, y; δ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , (x m , y m ) are drawn independently from a distribution supported on , where w ≤ W , and u : R → [0, 1] is a known nondecreasing 1-Lipschitz function.
Sentence 2: then for any δ ∈ (0, 1), the following holds with probability at least 1 − δ: there exists some iteration t < O(W m/ log(1/δ)) of GLM-TRON such that the hypothesis the theorem implies that some h t has small enough ε(h t ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Several recent papers [6,10,9,11] have considered the problem of recovering the connected components of {x : f (x) ≥ λ} for a user-specified λ: the flat version of our problem.
Sentence 2: the algorithm of [6] is intuitively similar to ours, though they use a single graph in which each point is connected to its k nearest neighbors, whereas we have a hierarchy of graphs in which each point is connected to other points at distance ≤ r (for various r).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let us define the weight of an example i under the current model as the rate at which loss decreases as the margin of i increases: (5) Now, to compute the gain (Equation 4), we note that: We recursively split leaf nodes by choosing the best atomic splitting feature that will allow us to increase the gain.
Sentence 2: we consider splitting each leaf node n using atomic feature â, where â = arg max Splitting using â would create children nodes n 1 and n 2 , with ϕ(n 1 ) = f ∧ â and ϕ(n 2 ) = f ∧ ¬â.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (Here, the expectation is with respect the randomization of the training set z 1 , .
Sentence 2: . . , z n used to produce w.) We make the following assumptions over the loss function f . First, we assume that for any fixed parameter z ∈ Z, the function f (•, z) is α-exp-concave over the domain W for some α > 0, namely, that the function exp (−αf (•, z)) is concave over W. We will also assume that f (•, z) is β-smooth over W with respect to Euclidean norm • 2 , which means that its gradient is β-Lipschitz with respect to the same norm: this property implies that f (•, z) is differentiable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, the dynamic programming framework [3] is employed to tackle the aforementioned issues.
Sentence 2: we first establish the fundamental theory of optimal campaigning over social networks where the user activities are modeled as a multivariate Hawkes process (MHP) [4,5] since MHP can capture both endogenous and exogenous event intensities.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, a crucial observation that we exploit in our work is that for observed data such as natural sensory input or data generated by a sparse coding model, the activity of latent components (or causes) can be expected to be concentrated in low-dimensional subspaces.
Sentence 2: for a given observed data point, all except for a very small fraction of the latent components can be assumed to be non-causal or irrelevant, hence the corresponding latent space can be neglected for the integration over � s. For a sparse initiation (i.e., π � 1) of the spike-and-slab model ( 1) to (2), we consider such low dimensional subspaces to be spanned by a few (approximately πH) of the H latent space coordinates.
Relationship: entailment
Domain: NeurIPS
Sentence 1: First, we assume that there exists some feasible parameter set Θ ⊂ R n that is known to contain the parameter θ * = (θ Acc , θ Sim ) that models the true distribution π * of the data: Next, we must be able to accurately learn θ * if we are provided with labeled samples of the true distribution.
Sentence 2: there must be an asymptotically unbiased estimator θ that takes some set of labeled data T independently sampled from π * such that for some c > 0, Finally, we must have enough sufficiently accurate heuristics so that we have a reasonable estimate of Y.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To accomplish this, we multiply w c i by H( wyi c − ), where H(•) is the Heaviside step function.
Sentence 2: we note that w c i refers to the weight of example x i to be borrowed by the target class c, whereas wyi c refers to the average weight of examples that class y i borrows from the target class c. if the examples that class y i borrows from class c have low weights on average (i.e.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let Φ ∈ R N H be such that its first d-coordinates are equal to C(n)U A S 1/2 A , and its remaining entries are 0.
Sentence 2: if ρ(n) = ω(log n) and K (F and κ) is well-conditioned, then with high probability: (2) by letting t = ρ 2/3 (n), we have Φ − Φ F = O √ nρ −2/87 (n) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 7 essentially implies that by exploiting the multi-stage convex relaxation framework, our proposed DC proximal Newton algorithm gradually reduces the estimation bias for "strong" signals, and eventually obtains an estimator with better statistical properties than the `1-regularized estimator.
Sentence 2: let e K be the smallest integer such that after e K stages of convex relaxation we have C 7 0.7 , which is equivalent to requiring e K = O(log log d).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We identify a greedy item placement strategy which outperforms all alternative approaches considered on real-world datasets, and can be proven to have a non-trivial recovery property: it recovers hidden co-clusters of items in probabilistic inputs subject to a recovery condition.
Sentence 2: we show that, given a set of hidden co-clusters to be placed onto k components, the greedy strategy will tend to place items from the same hidden cluster onto the same component, with high probability.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Those topics, as latent factors, can be interpreted by distributions over words and used to derive low dimensional representations of the documents.
Sentence 2: most existing topic models are built on top of the following generative process: Each topic is a distribution over the words (i.e., word distribution, WD) in the vocabulary; each document is associated with a topic proportion (TP) vector; and a word in a document is generated by first drawing a topic according to the document's TP, then sampling the word according to the topic's WD.
Relationship: entailment