sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We generalize the BMF framework by imposing that H is low-rank.
Sentence 2: we impose the rank-1 expansion H = Kc k=1 u :k v T :k , where u :k and v :k are column vectors (thus their outer product is a rank-1 matrix), each of them is modeled here by a Gaussian distribution: and K c is the number of such rank-1 matrices such that K c < min(K l , K r ), i.e., H is low-rank.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, we like to mention that, if the conditional distributions are not equal modulo translations, then the situation may become more involved.
Sentence 2: if we are in a situation with p(f * P (x)|x) > 0 and p(f (10) are in general impossible, and weaker self-calibration inequalities require additional assumptions on P. We refer to [8] where the case = 0 is considered.
Relationship: entailment
Domain: NeurIPS
Sentence 1: According to the above selection of hyperparameters on the synthetic datasets, we chose a wide range of hyperparameters to test KCDC on.
Sentence 2: both on the response and input we used either a log kernel k(x, x ) = − log( x − x q + 1) with q in [2, 3,4] or an RBF kernel with bandwidth [1, 1.5, 2] times the median heuristic.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , x p follow an iid uniform distribution on [2 for some > 0 then there exists a polynomial-in-p time algorithm which solves the subset-sum problem whp as p → +∞.
Sentence 2: even though the problem is NP-hard in the worst-case, assuming a quadratic-in-p number of bits for the coordinates of x, the algorithmic complexity of the typical such problem is polynomial in p. The successful efficient algorithm is based on an elegant application of a seminal algorithm in the computational study of lattices called the Lenstra-Lenstra-Lovasz (LLL) algorithm, introduced in Lenstra et al.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If K t = K for all t, then we say that we using a (nonadaptive) stationary controller K. In this setting, it is straightforward to check if our system is stable.
Sentence 2: it is BIBO stable if and only if the magnitude of all the eigenvalues of D = D t = A+BK t are strictly less than 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this section, we propose Influence Maximization Linear UCB (IMLinUCB), detailed in Algorithm 1.
Sentence 2: notice that IMLinUCB represents its past observations as a positive-definite matrix (Gram matrix) M t ∈ d×d and a vector B t ∈ d . let X t be a matrix whose rows are the feature vectors of all observed edges in t steps and Y t be a binary column vector encoding the realizations of all observed edges in t steps.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The expression in Theorem 3.1 can be simplified by examining its Taylor expansion (see Appendix A).
Sentence 2: for the special case of ✓ = 1 where the adversary establishes only one connection per server, line (5) simplifies to This suggests that the first-timestamp estimator has a probability of detection that decays to zero asymptotically as log(d)/d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To give more quantitative insight into these trends, we consider each of the six different experimental settings (i.e., (3 datasets) × (unsupervised vs. supervised)) as trials and consider what performance trends are likely to generalize.
Sentence 2: we use the non-parametric Wilcoxon Signed-Rank Test [33] to quantify the differences between the different aggregators across trials, reporting the T -statistic and p-value where applicable.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The explosion of data and an increase in model size has led to great interest in training deep neural networks on distributed systems.
Sentence 2: distributed stochastic gradient descent has been extensively studied in both deep learning and high-performance computing communities, with the goal of accelerating large-scale learning tasks [13,10,9,47,3,14,30,31,6,44,38].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Furthermore, every village j borrows only from those MFIs i ∈ B j that offer the lowest interest rate.
Sentence 2: i∈Bj ,r * i =r * m j x * j,i (1 + r * i − e j ) = d j for any MFI m j ∈ argmin i∈Bj r * i , and x * j,k = 0 for any MFI k such that r * k > r * mj .
Relationship: entailment
Domain: NeurIPS
Sentence 1: This leads to the cut functional cut CE , Note that in contrast to the hypergraph cut (1), the value of cut CE depends on the way each hyperedge is cut since the term |e ∩ C| |e ∩ C| makes the weights dependent on the partition.
Sentence 2: the smallest weight is attained if only a single vertex is split off, whereas the largest weight is attained if the partition of the hyperedge is most balanced.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Regularization has emerged as a dominant theme in machine learning and statistics, providing an intuitive and principled tool for learning from high-dimensional data.
Sentence 2: regularization by squared Euclidean norms or squared Hilbert norms has been thoroughly studied in various settings, leading to efficient practical algorithms based on linear algebra, and to very good theoretical understanding (see, e.g., [1,2]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper introduces SafetyNets, a new (and to the best of our knowledge, the first) approach for verifiable execution of deep neural networks on untrusted clouds.
Sentence 2: safetyNets composes a new, specialized IP protocol for the neural network's activation layers with Thaler's IP protocol for matrix multiplication to achieve end-to-end verifiability, dramatically reducing the bandwidth costs versus a naive solution that verifies the execution of each layer of the neural network separately.
Relationship: entailment
Domain: NeurIPS
Sentence 1: An important class of high order interactions occurs when the simultaneous effect of two variables on a third may not be additive.
Sentence 2: it may be possible that X ⊥ ⊥ Z and Y ⊥ ⊥ Z, whereas ¬ ((X, Y ) ⊥ ⊥ Z) (for example, neither adding sugar to coffee nor stirring the coffee individually have an effect on its sweetness but the joint presence of the two does).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given p agents of different radii {r i } p i=1 with given desired initial and final positions, {x i (0)} p i=1 and {x i (T )} p i=1 , along with a cost functional over trajectories, compute collision-free trajectories for all agents that minimize the cost functional.
Sentence 2: find a set of intermediate points {x i (t)} p i=1 , t 2 (0, T ), that satisfies the "hard" collision-free constraints that kx i (t) x j (t)k > r i + r j , for all i, j and t, and that insofar as possible, minimizes the cost functional.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The market maintains a market state q t ∈ R d at time t = 0, . . . , T , with q 0 = 0.
Sentence 2: each trader t = 1, . . . , T arrives sequentially and purchases a contract dq t ∈ R d , and the market state is updated to q t = q t−1 + dq t . q t = t s=1 dq s , the sum of all contracts purchased up to time t. The price paid by each participant is determined by a convex cost function C : R d → R. Intuitively, C maps q t to the total price paid by all agents so far, C(q t ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: It basically states that the Kantorovich metric is a solution to the Monge-Kantorovich problem, when its cost function is equal to the base metric for the Kantorovich metric.
Sentence 2: for two measures µ and ν, and a cost function f ∈ [S × S → R], the Monge-Kantorovich problem computes: The set of measures ξ with marginals µ and ν is also known as the set of couplings of µ and ν.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As the locality principle commonly exists in object-to-object relations in the real world, the tailor module enables the inference on the dynamics of objects focusing on the relations with neighbour objects.
Sentence 2: it crops a "horizon" window of size w from the object masks centered on the position of the dynamic object whose motion is being predicted, where w indicates the maximum effective range of object-to-object relations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We develop a framework which provides convergence conditions and error bounds along with simple proofs and connections to methods in numerical linear algebra.
Sentence 2: we show that if the Gaussian precision matrix is generalized diagonally dominant, then any Hogwild Gibbs sampler, with any update schedule or allocation of variables to processors, yields a stable sampling process with the correct sample mean.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To measure prediction performance, we use standard 5-fold cross-validation strategy by computing the root mean square error (RMSE) between the predicted and actual values of the cognitive scores on the testing data only.
Sentence 2: the whole set of subjects are equally and randomly partitioned into five subsets, and each time the subjects within one subset are selected as the testing samples and all other subjects in the remaining four subsets are used for training the regression models.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a phone sequence, we produce N -grams by the standard transformation of the stream; e.g., for bigrams (2-grams) the sequence of phones, t 1 , t 2 , ..., t n , is transformed to the sequence of bigrams of phones t 1 t 2 , t 2 t 3 , ..., t n−1 t n . We then find probabilities of Ngrams with N fixed.
Sentence 2: suppose we are considering unigrams and bigrams of phones, and the unique unigrams and bigrams are designated d 1 , ..., d M and d 1 d 1 , ... d M d M , respectively; then we calculate probabilities and joint probabilities where #(t k = d i ) indicates the number of phones in the conversation side equal to d i , and an analogous definition is used for bigrams.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Also of interest is the comparison with bounds for low-rank matrices, for which X Σ ≤ √ rank X X Fro .
Sentence 2: for n × m rank-k X with entries bounded by B, X Σ ≤ √ knmB, and the second term in the right-hand side of (7) becomes: Although this is the best (up to log factors) that can be expected from scale-sensitive bounds 4 , taking a combinatorial approach, the dependence on the magnitude of the entries in X (and the margin) can be avoided [9].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead of computing λ k+1 exactly according to (13), we propose to approximate λ k+1 with the help of Lemma 3.1 and set µ k+1 to this approximation.
Sentence 2: let t k be the total number of consensus steps done before k-th iteration of PDA, and let q k ≥ 1 be the number of consensus steps within iteration k. to approximate P C (x) in (13).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unfortunately, while various such multiplicative weights have been designed for voting with subtrees [3,4,5], they are not known to yield statistically consistent prediction.
Sentence 2: in fact, the best known result to date [5] presents a weighting scheme which can provably achieve an excess risk 3 (over the Bayes classifier) of the form o P (1) + C • min T R(h T ), where R(h T ) denotes the misclassification rate of a classifier h T based on subtree T . the excess risk might never go to 0 as sample size increases, which in contrast is a basic property of the pruning alternative.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Likewise we may perform the linearization in (6) as follows: In other words, we use the rescaled estimate ỹ to provide an upper bound on the concave part of the loss function.
Sentence 2: this leads to the following instantiation of standard convex-concave procedure: instead of the structured estimation loss it uses the loss bound l(x, y, ỹ, f ) In the case of Γ(η) = 1 this can be simplified significantly: the terms in f (x, y) cancel and l becomes we replace the correct label y by the rescaled estimate ỹ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then for any f ∈ C s 0 ∩ W s 2 with s ≤ r, we have Consequences similar to those of Theorems 1 and 2 can be drawn for Theorem 3.
Sentence 2: the rate in (16) coincides with that of (15) with c = 1/2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: On the other hand, the supervised FSDR approaches include supervised principal component analysis [9], sliced inverse regression [10], and kernel dimension reduction [11].
Sentence 2: for multi-label classification, a leading supervised FSDR approach is canonical correlation analysis [CCA; 6,12] which is based on linear projections in both the feature space and the label space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We also demonstrate empirically that our algorithms outperform existing methods, such as those of [5,18,14], for ARMP and the matrix completion problem by an order of magnitude and are also more robust to noise and sampling schemes.
Sentence 2: results show that our SVP-Newton method is significantly robust to noise and performs impressively on a more realistic power-law sampling scheme for the matrix completion problem.
Relationship: entailment
Domain: NeurIPS
Sentence 1: V ar(â vw,s ) = (s − 1) Interestingly, the variance (15) says we do need s = 1, otherwise the additional term (s − 1) D i=1 u 2 1,i u 2 2,i will not vanish even as the sample size k → ∞.
Sentence 2: the choice of random distribution in VW is essentially the only option if we want to remove the bias by premultiplying the data vectors (element-wise) with a vector of random variables.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Connections to other DP-based topic models PFA is a nonnegative matrix factorization model with Poisson link that is closely related to other DP-based models.
Sentence 2: [32] showed that by making p(h kn = 1) = 1 and letting θ kn have a Dirichlet, instead of a Gamma distribution as in (1), we can recover LDA by using the equivalence between Poisson and multinomial distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The notation (R x y) means that x and y are related by R. In miniKanren queries, data flow is not directionally biased: any input to a relation can be unknown.
Sentence 2: for example, a query (R X y) where y is known and X is an unknown, called a logic variable, finds values X where X and y are related by R. given R and f defined as before, the query finds inputs X to f such that (f X) = y.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our offline version of UCSG, in each phase k we let both players each select their own optimistic policy.
Sentence 2: after Player 1 has optimistically selected π 1 k , Player 2 then optimistically selects his policy π 2 k based on the known π 1 k . the model-policy pair M 2 k , π 2 k is obtained by another extended value iteration on the extended MDP under fixed π 1 k , where Player 2's action set is extended.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This is meaningful because in most applications, the time complexities for evaluating gradients at different points are of the same magnitude.
Sentence 2: the worse-case time complexities of first-order methods are usually proportional to a fixed parameter times T . In large-scale settings, however, if we have already spent time computing the (full) gradient at x, perhaps we can use such information to reduce the time complexity to compute full gradients at other points near x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The second term indicates the regularization effects due to the max-margin posterior constraints, which is consistent with our intuition.
Sentence 2: for those data with non-zero Lagrange multipliers (i.e., the data are around the decision boundary or misclassified), the second term will bias the model towards a new posterior distribution that favors more discriminative representations on these "hard" data points.
Relationship: entailment
Domain: NeurIPS
Sentence 1: PGRD cheaply computes the gradient of the objective utility with respect to the R i parameters through UCT planning.
Sentence 2: it takes a simulation model of the CMP and an objective reward function and uses UCT to simultaneously plan actions with respect to the current internal reward function as well as to update the internal reward function in the direction of the gradient of the objective utility for use in the next planning step.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In this paper, we attempt to address two challenges: 1) to break away from the standardized norm of feature-based deep transfer learning 1 , and 2) to learn versatile structures in the data with a data-driven approach.
Sentence 2: we are interested in learning transferable latent relational graphs, where the nodes of a latent graph are the input units, e.g., all the words in a sentence.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Given that the discriminator class is sufficiently large, [9] interpreted the GAN training as finding a generator such that the generated distribution ν is as close as possible to the target true distribution µ, measured by the Jensen-Shannon distance d JS (µ, ν), as shown below: Inspired by such an idea, a large body of GAN models were then proposed based on various distance metrics between a pair of distributions, in order to improve the training stability and performance, e.g., [2,3,13,15].
Sentence 2: among them, the integral probability metric (IPM) [19] arises as an important class of distance metrics for training GANs, which takes the following form different choices of the function class F in (2) result in different distance metrics.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, the meaning of this near optimality of g * ι,θ (s, a; ω) is interpreted as a near minimization of the Euclidean distance between r and its approximator R ρ * (s, a; ω), so that ρ * works to reduce the distance of the regressand r and the subspace of the regressor ∇ θ ln{d θ (s) ι π(a|s;θ)} of the M-estimator ω.
Sentence 2: r(s, a) is almost in this subspace at the point ω = ω * .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Optimizing M k . First we fix w j k to optimize M k . To avoid over-fitting, as in support vector machine (SVM), we minimize the overall C2B distances from X i 's associated classes to itself and the total amount of slack.
Sentence 2: we solve the following convex optimization problem: where C is a trade-off parameter, acting same as in SVM.
Relationship: entailment
Domain: NeurIPS
Sentence 1: W e show in this paper that mainstream inference algorithms based on jointrees [14,8] are a special-case of the arithmetic-circuit approach proposed in [2].
Sentence 2: we show that each jointree is an implicit representation of an arithmetic circuit; that the inward-pass in jointree propagation ev aluates this circuit; and that the outwardpass differentiates the circuit.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We follow the three steps in Section 4 to develop MLFre.
Sentence 2: we first present an accurate estimation of the dual optimum in Section 5.1, then we solve for the supreme values in (R1 * ) and (R2 * ) in Section 5.2, and finally we present the proposed MLFre in Section 5.3.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The idea of MMAX is to consider a modular lower bound tight at the set corresponding to each block of a partition.
Sentence 2: at iteration t + 1, for each block i, we approximate f i with its modular lower bound tight at A π t i and solve a modular version of Problem 1 (Line 7), which admits efficient approximation algorithms [2].
Relationship: entailment
Domain: NeurIPS
Sentence 1: In machine learning, the properties of k-means have been studied, for fixed k, by considering the excess reconstruction error E ρ (S n,k ) − E ρ (S k ).
Sentence 2: this quantity has been studied for X = R d , and shown to be, with high probability, of order kd/n, up-to logarithmic factors [31].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, let E be drawn uniformly from I(x), so that P is extended to be a joint distribution over (x, y, E).
Sentence 2: we say that a forecaster averaged over all x, y and events of interest E ∈ I(x), whenever the forecaster outputs probability p, then the event E actually holds with probability p. Note that this definition corresponds to perfect binary calibration (1) for the transformed pair of variables y = I[y ∈ E], x = (x, E).
Relationship: entailment
Domain: NeurIPS
Sentence 1: A Central Limit argument appeals to the number of independent eigenfunctions that contribute to this residual, and, the orthogonality of eigenfunctions implies that as more of them meaningfully contribute to each entry in the residual, the more independent those entries become.
Sentence 2: if this long tail begins at a low magnitude and decays slowly, then we can treat it as a Gaussian i.i.d.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To learn Q automatically, we propose an attention model which guides the random surfer on "where to attend to" as a function of distance from the source node.
Sentence 2: we define a Graph Attention Model as a process which models a node's context distribution Q as the output of softmax: where the variables q k are trained via backpropagation, jointly while learning node embeddings.
Relationship: entailment
Domain: NeurIPS
Sentence 1: . . , (x t , y t )}, denote the current version space by V (S t ) = {h ∈ H| S | ∀(x, y) ∈ S t , h(x) = y}.
Sentence 2: the next query selected by the algorithm is the algorithm selects the query that in the worst-case over the possible hypotheses, would remove the most hypotheses from the version spaces, when normalizing by the outcome-dependent cost of the query.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our initial experiments, we found that the use of a single state to represent all the regions of baseline in the ECG waveform resulted in poor performance when the model was used to infer the underlying state sequence of new unseen waveforms.
Sentence 2: a single baseline state allowed for the possibility of the model returning to the P wave state, following a P wave -Baseline sequence.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The new posterior approximation takes the same form, but with the new pseudo-points and new hyperparameters: Similar to the batch case, this approximate inference problem can be turned into an optimisation problem using variational inference.
Sentence 2: consider Since the KL divergence is non-negative, the second term in the expression above is the negative approximate lower bound of the online log marginal likelihood (as Z 2 /Z 1 ≈ p(y new |y old )), or the variational free energy F(q new (f ), θ new ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Garivier and Kaufmann [8] recently provided a general lower bound on the number of samples required in the fixed confidence setting.
Sentence 2: they show that for any normal bandit model, under any sampling rule and stopping time τ δ that guarantees a probability of error no more than δ, Recall that M β , defined in (3), is the first time after which the empirical proportions are within of their asymptotic limits.
Relationship: entailment
Domain: NeurIPS
Sentence 1: While classical subgradient methods only achieve an error level on the objective function of O(1/ √ k) after k iterations, proximal-gradient methods have an error of O(1/k) while accelerated proximal-gradient methods futher reduce this to O(1/k 2 ) [1,2].
Sentence 2: accelerated proximalgradient methods for non-smooth convex optimization achieve the same optimal convergence rate that accelerated gradient methods achieve for smooth optimization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Unfortunately, inference and learning in this network (and in general in coupled Markov chains) requires very expensive computations, and cannot be performed by simply propagating messages over time and among the variable nodes.
Sentence 2: the state of a child variable x k t depends on x k t−dt , s k t , x 1 t and the state of all other children at the previous time step, [x j t−dt ] 2<j<n,j =i .
Relationship: entailment