sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Note that draws from X must be consistent with the observations Y 1:n . The usual approach to the consistency issue is to make a proposal by conditioning a linear diffusion to hit some neighbourhood of the observation Y k , then to make a correction via a rejection sampling [18] or a Metropolis-Hastings [16] step.
Sentence 2: in our experiments, the algorithms terminated in a roughly similar length of time (though no serious attempt was made to optimise the runtime of either method).
Relationship: neutral
Domain: NeurIPS
Sentence 1: More efficient techniques could also be employed (including several proposed for CS in the presence of noise [3,5,7,10,12]).
Sentence 2: in addition, given the (K + K 1 )c measurements for x 1 as side information, and assuming that the partitioning of x 1 into z C and z 1 is known, cK 2 measurements that describe z 2 should allow reconstruction of x 2 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: 6, it can be inferred that the holistic feature vector X can be obtained by X = X − β ∂L ∂X , where β = α Z 2 , implying that updating weight parameters w with SGD can implicitly lead to the update of the output feature in the similar way.
Sentence 2: considering the distribution symmetry of W , the angle range of the constrained region for class y i is 2θ = 2π c , which is inversely proportional to c. if the class number increases, the constrained region of each class will be more and more compact.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that θ corresponds to the adjacency matrix constructed by the p components of X as nodes, and θ ij = 0 indicates that there is an edge between X i and X j .
Sentence 2: 2011, Mazumder and In the Gaussian case, the screening rule can be attained by simply replacing the second empirical moment matrix in (4) with the sample Algorithm 1 Blockwise Minimization 1: Input: dataset X, regularization parameter λ.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For dense graphs satisfying d = Ω(n) (which is studied in [FSGC14]), we prove a stronger upper bound of α(n) = O( 1 √ n ) using the Szemerédi regularity lemma.
Sentence 2: we show that this naive bound can be asymptotically improved upon, by constructing an α-multitaskers with α = Ω( log d d ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We call these experiments because the goal of solving these problems is to disambiguate nondeterministic action models.
Sentence 2: a set of goals G is a cover of a set of states R if g∈G {s : s satisfies g} = R. Given the goal language G and a model M , the problem of experiment design is to find a set of goals G ⊆ G such that the sets of states that satisfy the goals in G collectively cover all informative states I(M ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: As another useful property of LARS/Homotopy algorithms, it is worthwhile to mention that changes of the active set only happen at the breakpoints [17,7,15].
Sentence 2: c) The Alzheimer's disease neuroimaging initiative (ADNI; available at www.loni.ucla.edu/ADNI) studies the disease progression of Alzheimer's.
Relationship: neutral
Domain: NeurIPS
Sentence 1: When written in matrix form, the ADCLUS model seeks to uncover a feature matrix F and a weight vector w such that S ≈ FWF , where W = diag(w) is a diagonal matrix with nonzero elements corresponding to the saliency weights.
Sentence 2: since the IBP describes a prior over infinite feature matrices, the resampling procedure needs to accommodate the remaining (infinite) set of features that are not currently represented among the manifest features F. When resampling feature assignments, some finite number of those currently-latent features will become manifest.
Relationship: neutral
Domain: NeurIPS
Sentence 1: When the training distribution is sufficiently close (top row), the modes "snap" together and the mean of the two modes is assigned high probability.
Sentence 2: finally for multiple features, when the training set only contains certain combinations, e.g., red cubes but not yellow cubes (Figure 1,D), we find that the learning algorithms will memorize the combinations in the training set when it contains a small number of them (e.g., 20), and will generate new combinations (not in the training set) when there is more variety (e.g., 80).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The signal-to-noise ratio (SNR) is defined as 10 log 10 (E s /N 0 ) where E s /N 0 = ne σ 2 (n is the number of variables, σ 2 is the variance of the Gaussian additive noise, and e is the variance of the uniform distribution over the discrete set A).
Sentence 2: the proposed variant of the BP algorithm is both computationally efficient and achieves near optimal results.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We start by defining the notion of approximate coverage, which is different than the corresponding notion in [8] for p ≥ 1, due to the fact that 0 -regression cannot be efficiently solved.
Sentence 2: given a matrix A ∈ {0, 1} m×n , if OPT A / A 0 ≤ 1/240 then we can exactly solve the Boolean 0 -rank-1 problem in time 2 O(OPT/ √ A 0) • poly(mn).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The KPCA procedure approximates this objective by considering the empirical covariance operator, denoted C n , and the subspace S D spanned by its first D eigenvectors.
Sentence 2: this does not tell us much about the convergence of S D to S D -since two very different subspaces can have a very similar reconstruction error, in particular when some eigenvalues are very close to each other (the gap between the eigenvalues will actually appear as a central point of the analysis to come).
Relationship: neutral
Domain: NeurIPS
Sentence 1: They are employed to assess different aspects of a given clustering: PUR measures the accuracy of the dominating class in each cluster; NMI is from the information-theoretic perspective and calculates the mutual dependence of the predicted clustering and the ground-truth partitions; RI evaluates true positives within clusters and true negatives between clusters; and FM considers both precision and recall.
Sentence 2: moreover, the first constraint in Eq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: First, given some region S, we must calculate an upper bound on the scores D(S ) for regions S ⊂ S. More precisely, we are interested in two upper bounds: a bound on the score of all subregions S ⊂ S, and a bound on the score of the outer subregions of S (those regions contained in S and containing its center S C ).
Sentence 2: computing the scan statistic for all spatial regions is generally computationally infeasible, so we introduce a novel fast spatial scan algorithm, generalizing the 2D scan algorithm of (Neill and Moore, 2004) to arbitrary dimensions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If (x, z) = e ihz,xi ˆ (z) 1/2 and ⌫ / ˆ for 2 L 2 , then R SD ,2,⌫,M is the random Fourier feature (RFF) approximation [22] to KSD k with k(x, y) = (x y).
Sentence 2: our next result, proved in Appendix B, shows that tilted base kernels with A increasing sufficiently quickly also control convergence.
Relationship: neutral
Domain: NeurIPS
Sentence 1: . . , S k } to minimize the sum k i=1 f (S i ), then we have an exact algorithm for 2-clusterings and a factor 2 approximation guarantee.
Sentence 2: we show that the optimal partitioning into 2 clusters, and approximate partitioning (guaranteed to be within a factor of 2 of the the optimal) for more clusters can be computed.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since the µ ℓ are strictly decreasing, we conclude that β ∈ O S . Moreover, choosing q ∈ J ℓ and summing both sides of equations (A.5) over j ∈ J ℓ,q we obtain that which implies that β ∈ Q J ℓ ,J ℓ,q .
Sentence 2: the last claim in the theorem follows from the fact that the set {γ(ǫ) : ǫ > 0} is bounded and the function λ(β) is continuous.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Tracing its origins to [17] for offline optimization problems, OMD proceeds by taking a gradient step in the dual (gradient) space and projecting it back to the primal (decision) space via a mirror map generated by a strongly convex regularizer function (with different regularizers giving rise to different algorithms).
Sentence 2: if C is nonempty and λ-stable, then it is closed, convex and contains all Nash equilibria of the game.
Relationship: neutral
Domain: NeurIPS
Sentence 1: < 2.373 be the exponent of fast matrix multiplication, the algorithm runs in We note that the runtime of our algorithm is O(nnz(A)) whenever n, k, 1/ k+1 , and 1/✏ are not too large.
Sentence 2: kernel methods typically work with the kernel matrix K 2 R n⇥n with K i,j = (a i , a j ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Local Gaussian Smoothing As another approach to increase smoothness, the blockwise uniform distribution in (5) can be replaced by a Gaussian distribution g k = N (θ; µ k , Σ k ), with mean and covariance estimated "locally" by samples within the block.
Sentence 2: as both are kernel-based, the two methods are limited by two major drawbacks.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Critchley (1978) [3] (also mentioned in section 2.4.2 of Cox and Cox) carried out metric MDS by running the classical scaling algorithm on the transformed dissimilarities.
Sentence 2: we can again show that the kernel MDS procedure operates on the matrix H K H. the distance 8ij in feature space is not a function of 8ij and so the relationship of equation 3 does not hold.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The learner also assumes that the reward function is linear, but in features which are different from the truly relevant ones; e.g., the learner could only observe a subset of those features.
Sentence 2: yet another way of characterizing the teaching risk is as ρ(A L ; w * ) = pr(w * ) , where pr : R k → ker A L denotes the orthogonal projection onto ker A L . Significance of teaching risk.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If we let Φ t = G(x t ) (assuming for sake of clarity that x t > 0), where Since the payoff of the algorithm at time step t is g(x t )b t , we have where Z is the desired bound on per step loss of the algorithm.
Sentence 2: an important property of our algorithm is that it does not need a high imbalance between the number of ones and the number of zeros in the whole sequence to have a gain: it is sufficient for the imbalance to be large enough in at least one contiguous time window 2 , the size of which is a parameter of the algorithm.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We present the following two realizations of this operator that provides the output of a set of filters in a neighborhood of a node of interest that we refer to as the "center node": where N i is the set of neighbors of node i, W C is the weight matrix associated with the center node, W N is the weight matrix associated with neighboring nodes, and b is a vector of biases, one for each filter.
Sentence 2: note that DCNNs use residue distances to inform the diffusion process.
Relationship: neutral
Domain: NeurIPS
Sentence 1: An MDP is weakly communicating (or weak accessible) if its states can be partitioned into two subsets: in the first subset all states are transient under every stationary policy, and every two states in the second subset can be reached from each other under some stationary policy.
Sentence 2: the above methods to end an episode can not be applied to MDPs without the special features.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In fact, the volume measure VM is related to the number of "distinguishable" probability distributions indexed by the model M.l Because of the way the model family is embedded in the space of distributions, two different parameter values can index very similar distributions.
Sentence 2: u sing Bayes rule, the probability that the truth lies in the family f given the observed data y can be written as: Here wee) is the prior probability of the parameter e, and A(f, y) = Pr(f)/Pr(y) is the ratio of the prior probabilities of the family f and data y. Bayesian methods assume that the latter are the same for all models under consideration and analyze the socalled Bayesian posterior Lacking prior knowledge, w should be chosen to weight all distinguishable distributions in the family equally.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Tarone's testability criterion has only been successfully applied to unconditional association tests, such as Fisher's exact test [6] or Pearson's χ 2 test [11].
Sentence 2: [18]), or (b) using Tarone's testability criterion, losing the ability to account for covariates and resulting in potentially many confounded patterns being deemed significant [15,13,7,8].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then, by Lemma 3.3 and the fact that R1,...,R K (X | S1,...,S K ) = R1,...,R K (π(X | S1,...,S K )), we have ) be as in the statement of Lemma 3.3.
Sentence 2: if the selected Tucker rank is too large, the computational cost of computing the Tucker decomposition (even if we allow for approximation methods) increases considerably along with space usage.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We introduce a natural model that scales between A-and D-optimal design.
Sentence 2: our motivation here is the dual problem of D-optimal design (i.e., dual to the convex relaxation of D-optimal design): this is nothing but the well-known Minimum Volume Covering Ellipsoid (MVCE) problem, which is a problem of great interest to the optimization community in its own right-see the recent book [42] for an excellent account.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Second, existing work on this topic has so far not provided any framework capable of addressing what is perhaps the most basic question about δ-sampling: given a tree T δ with its set of exposed nodes indicated -i.e., a single outcome of the δ-sampling process -can we infer the number of nodes in the original tree T ?
Sentence 2: the key to the argument is Theorem 2.1, which asserts that with high probability, T δ has Ω(δn log k δ −1 ) nodes in total.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Solving this transport equation on [0, t] essentially transports the initial mass to the mass at time t. Algorithm 2: HYBRID MASS TRANSPORT Input: Sample size n, time t, ⌧ Output: μn(t), ˆ n(x, t) Generate n samples of point process: To efficiently solve the mass transport equation, we reformulate it as a banded linear ODE.
Sentence 2: let (t) := (t|H t ) be the conditional intensity function of the point process N (t) and ˜ (x, t) := P[N (t) = x|H t ] be its conditional probability mass function; then ˜ (x, t) satisfies the following differential-difference equation: loss in mass, at rate (t) Proof sketch.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The concept class C induced by a learning algorithm L : ∪ ∞ m=1 Z m → H is defined to be the set of total Boolean functions c(z) = I{P(Z trn = z | H) ≥ P(Z trn = z)} for all H ∈ H. Intuitively, every hypothesis H ∈ H induces a total partition on the observation space Z given by the Boolean function in Definition 8.
Sentence 2: also, given two probability mass functions P and Q defined on the same alphabet A, we will write P, Q to denote the overlapping coefficient, i.e.
Relationship: neutral
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: this work was partially supported by U.S. NSF-IIS 1302675, NSF-IIS 1344152, NSF-DBI 1356628, NSF-IIS 1619308, NSF-IIS 1633753, NIH AG049371.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our results show that, even on a single machine, FALKON can outperforms state of the art methods on most problems both in terms of time efficiency and prediction accuracy.
Sentence 2: for example, if a simple gradient descent ( 6) is used, the number of iterations needed for an accurate solution of problem ( 5) is t = O(cond(K nn + λnI) log(1/ )).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We start by defining the following terms for the independent cascade model IC(G, P, S 0 ), where , the set P l (S 0 → v) is defined as the set of all paths with length l from any seed s ∈ S 0 to v. We call a path P is open iff every edge in P is open.
Sentence 2: initial condition: For every s ∈ S 0 , s + ∈ N + (s), u ∈ V \S 0 , and v ∈ N + (u), Recursion: Equation ( 10) follows from that for any seed node s ∈ S 0 and for all l > 0, the probabilities p l (s) = 0, p l (s → s + ) = 0, and p l (s , but this results in an extremely loose bound due to the backtracking.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Finally, when the dataset satisfies certain regularity conditions, the size of I can be improved, which reduces both embedding and clustering runtime.
Sentence 2: space H n if Notice that the inverse since φ i ∈ Φ n , we have that in the worst case the error is bounded as we know that in the best case the error can be reduced up to Note that the directions associated with the larger eigenvalues are the ones that occur most frequently in the data.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Form the second order approximation fXt (∆) := ḡXt (∆) + h(X t + ∆) to f (X t + ∆).
Sentence 2: at most linear rates in turn become infeasible when the problem size is very large, with the number of nodes in the thousands and the number of matrix entries to be estimated in the millions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: From these plots, we notice that the risk-sensitive algorithms that we propose result in a long-term (discounted or average) reward that is higher than their riskneutral variants.
Sentence 2: from the Bellman equation of Λ µ (x), proposed by Sobel [19], it is straightforward to derive Bellman equations for U µ (x) and the square reward action-value function [17]).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We restrict interest here to the case of simple linear decoding functions Since p(r|s) is Gaussian, (2s i − 1)f i (r) ≡ x i is also Gaussian, In general, the average over the factorised distribution p(s) can be evaluated by using the Fourier Transform [1].
Sentence 2: we have described a general theoretically justified approach to information maximization in noisy channels.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In distributed environments without a high rate of connectivity, the decreased network usage of the mixture weight method should lead to substantial gains in wall clock as well.
Sentence 2: the application of the Cauchy-Schwarz inequality to the inequality just established yields The gradient of w → L zm (w) = log y∈Y e w•Φ(xm,y) −w • Φ(x m , y m ) is given by ∇L zm (w) = y∈Y e w•Φ(xm,y) Φ(x m , y) we obtain ∇L zm (w ) ≤ E y∼p w [•|xm] Φ(x m , y) − Φ(x m , y m ) ≤ 2R and similarly ∇L z m (w) ≤ 2R, which leads to the statement of the theorem.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For each of these 147 e-mails, we counted the number of e-mail addresses to which the message was sent, including both direct and CCed recipients.
Sentence 2: theorem 2.1 provided the main step; from here, we simply argue, in Theorem 2.2, that T δ will have at most O(δn) branching nodes with high probability.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To conclude, the left and right eigenvectors of the finite matrix M can be viewed as discrete approximations to those of the operators T f and T b , which in turn can be viewed as approximations to those of H f and H b .
Sentence 2: the term ∇ψ • ∇U in ( 17) is interpreted as a drift term towards low energy (high-density) regions, and as discussed in the next section, may play a crucial part in the definition of clusters.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We support our method with both theoretical and empirical analyses.
Sentence 2: the Curse of Horizon The importance weight w 0:T is a product of T density ratios, whose variance can grow exponentially with T . IS-based estimators have not been widely successful in longhorizon problems, let alone infinite-horizon ones where w 0:1 may not even be well-defined.
Relationship: neutral
Domain: NeurIPS
Sentence 1: More recently, [12] proposes an approximate objective for structural SVMs that leads to an algorithm considerably faster than DLPW on problems requiring expensive inference.
Sentence 2: in practice, since the set of active labels is not changing frequently during training, once an active label y f is sampled, it will be kept in the active set A f till the end of the algorithm and thus results in a convergence rate similar to that of an exact FMO.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Assume that The computational complexity of n FSSD 2 and σ2 H1 is O(d 2 Jn).
Sentence 2: such tests can require space partitioning [18,3], which works poorly in high dimensions; or closed-form integrals under the model, which may be difficult to obtain, besides in certain special cases [2,5,30,26].
Relationship: neutral
Domain: NeurIPS
Sentence 1: In fact, our method guarantees a nearly optimal error rate of It is noteworthy that CRR can tolerate a constant fraction of corruptions i.e.
Sentence 2: also let the number of corruptions k * be s. The above result establishes consistency of the CRR method with an error rate of Õ(σ d/n) that is known to be statistically optimal.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Because a sphere is rotationally symmetric, our forward model is insensitive to orientation, and similarly the external magnetic field caused by a dipole in a homogeneous sphere is invariant to the sphere's radius.
Sentence 2: (2002), except that an additional input was associated with each exemplar, namely the (x, y, z) coordinates of the center of a sphere fitted to the head, and the forward model was modified to account for this offset.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the case of imperfectinformation extensive-form games, one limitation is that some NEs do not prescribe optimal play after the player or the opponent has made a mistake.
Sentence 2: there exists a ¯ > 0 such that t i ( ) has the same sign as the first non-zero derivative of t i evaluated in 0 for all 0 < < ¯ .
Relationship: neutral
Domain: NeurIPS
Sentence 1: By considering a cost function c : Ω S × Ω T → [0, ∞[, Monge's formulation of the OT problem is to find a transport map T : Ω S → Ω T (also known as a push-forward operator) between two probability measures µ S on Ω S and µ T on Ω T realizing the infimum of the following function: When reaching this infimum, the corresponding map T is an optimal transport map.
Sentence 2: as such, our technique can readily learn the transfer function T s→t over a limited set of gradients and generalizes appropriately to unseen gradients.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Prior work has considered strongly convex decision sets [14], yet with the additional assumption that the objective is smooth and strongly convex, leading to O(1/T 2 ) convergence.
Sentence 2: if the constraint set is strongly convex, then this exhibits a novel projection free algorithm that grants a O(log T /T ) convergence even for non-smooth objective functions.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We prove upper bounds on the convergence rate of the classification error towards the Bayes risk for a variety of functions φ and of distributions P , in particular for SVM (Theorem 6).
Sentence 2: these results are particularly relevant to the one-class SVM, for which the regularization can not vanish by construction, and which is shown for the first time to be a consistent density level set estimator.
Relationship: neutral