sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: In this subsection, we present how geodesics in the Poincaré ball model are usually described with Möbius operations, and push one step further the existing connection between gyrovector spaces and the Poincaré ball by finding new identities involving the exponential map, and parallel transport.
Sentence 2: these findings provide us with a simpler formulation of Möbius scalar multiplication, yielding a natural definition of matrix-vector multiplication in the Poincaré ball.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a joint CDF, the problems of inference and likelihood evaluation, or computing conditional CDFs and marginal PDFs, both correspond to mixed differentiation of the joint CDF [6].
Sentence 2: the conditional CDF (x |x ) is related to the mixed derivative ∂ In the case of evaluating the likelihood corresponding to the model, we note that for CDF (x| ), the PDF is defined as In order to perform maximumlikelihood estimation, we require the gradient vector ∇ log (x| ) =  As described above, inference and learning in a CDN corresponds to computing derivatives of the CDF with respect to subsets of variables and/or model parameters.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 16 The modeling studies are also more ambiguous than they might seem, since their most convincing focus is on the tribulations of catastrophic interference.
Sentence 2: 17 slow learning is necessary in systems with rich distributed or population coding because changes in synaptic efficacies occasioned by incorporating new information can easily overwrite the neural substrate for the storage of old information (the hoary stability-plasticity dilemma 18 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: ⇤ In this section, we propose three algorithms that can be applied to solve the KCMAB problem and present their analyses.
Sentence 2: we consider the Upper Confidence Bound (UCB) Policy [2], and Choose arm a(t) = t. 3: end for 4: For all arm i, compute r i (t) = q 2 log t Ni(t) and u i (t) = μi (t) + r i (t) Choose arm a(t) = argmax i u i (t) (with compensation max j μj (t) μa(t) (t)) 7: end for Algorithm 1: The UCB algorithm for KCMAB.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Specifically, the reward is determined by values of: (1) the UCs governing the MUCT; (2) a set of unobserved variables (other than the UCs) where each affects an endogenous variable in the MUCT; and (3) the IB.
Sentence 2: there is no UC interplaying across MUCT and its outside so that µ x = E[Y |x] where x is a value assigned to the IB X.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This situation poses difficulties for some existing separation and localization algorithms, which often assume large numbers of microphones [Dorfan et al., 2015], an array of closely spaced microphones [Mandel et al., 2009, Lewis, 2012, or known sound characteristics [Oldfield et al., 2015, Wang andChen, 2017].
Sentence 2: we examine the situation of sound sources in a large, known, anechoic two-dimensional space, with pairs of two omni-directional microphones placed at arbitrary points and orientations.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Algorithm  with equality if X ⊤ S X S ≻ 0 for all subsets S of size k. Lemma 4.2 extends a result from [2, Lemma 3.9] on column-subset selection via volume sampling to all ESPs.
Sentence 2: it follows that removing one element (by volume sampling a set of size n − 1) will in expectation decrease f by a multiplicative factor which is clearly also attained by a greedy minimization.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We model the locations for all ROIs in the atlas space using a Gaussian model (mean: 𝑀 𝑋 𝑗 ,and standard deviation:  Under the above definition, if any ij X is within the range of 3 j X s from the distribution model center j X M , the anatomical constraint energy will always be one; if not, there will be an exponential increase of the energy which punishes the possible involvement of outliers.
Sentence 2: this energy factor will ensure the optimized ROIs will not significantly deviate away from the original ROIs.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In our setting, input bits are deterministic, hence independent.
Sentence 2: mathematically, a protocol P is protocol compatible if each column P (τ ) is a rank-one tensor, when reshaped into a k-th order tensor P (τ ) ∈ [0, 1] 2×2×...×2 , where there exist vectors u (1) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that a product distribution is one where all marginals are independent, so this is equivalent to testing if a collection of random variables are all independent.
Sentence 2: in the case where n `'s are large, then the first term dominates, and the sample complexity is O(( when d is a constant and all n `'s are equal to n, we achieve the optimal sample complexity of ⇥(n d/2 /" 2 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: In [19], an order between the concentrability coefficients was introduced: a coefficient A is said to be strictly better than B -a relation we denote with A ≺ B -if and only if i) B < ∞ implies A < ∞ and ii) there exists an MDP for which A < ∞ and B = ∞.
Sentence 2: it was argued that In this sense, 1) (µ, ν), while its definition might suggest improvement as κ increases.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The canonical aspect ratios can be determined by clustering aspect ratios of all ground-truth bounding boxes [22], or based on the viewpoints, provided we have labels for each viewpoint.
Sentence 2: suppose that there is a candidate example x i to be borrowed by the target class c and there are L canonical aspect ratios of c. We transform x i into x l i by resizing one dimension so that {x l i } 0≤l≤L contains all L canonical aspect ratios of c (and x 0 i = x i ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Yet another approach (if applicable) is to use a "comple-mentary mixtures of Z MLP-experts" model (or a neuro-fuzzy modular network) that combines Z smaller-size MLPs complementarily; the associated residual vector to be minimized becomes: r(") = y(") − t = Z i=1 wioi − t, where scalar w i , the ith output of the integrating unit, is the ith (normalized) mixing proportion assigned to the outputs (F -vector o i ) of expert-MLP i.
Sentence 2: note that each expert learns "residuals" rather than "desired outputs" (unlike in the committee method below) in the sense that only the final combined outputs y must come close to the desired ones t. there are strong coupling effects (see page 80 in [5]) among all experts; hence, it is crucial to consider the global Hessian across all experts to optimize them simultaneously [7].
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note also that our analysis applies to more adversarial settings than stochastic optimization (e.g., to online convex optimization [5]).
Sentence 2: an adversary may choose an arbitrary sequence of functions subject to the random data sparsity constraint (2), and our results provide an expected regret bound, which is strictly stronger than the stochastic convergence guarantees provided (and guarantees high-probability convergence in stochastic settings [3]).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We note that while the 2 statistic for testing hypothesis is prevalent in statistics providing optimal error exponents in the large-sample regime, to the best of our knowledge, in the small-sample regime, modified-versions of the 2 statistic have only been recently used for closeness-testing in [19,20,21] and for testing uniformity of monotone distributions in [22].
Sentence 2: [19] design an unbiased statistic for estimating the 2 distance between two unknown distributions.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In addition, we found out experimentally that when the data contains object images with rich backgrounds, performance of subclass recognition and localization is improved when using models with increased relative location weight.
Sentence 2: a part hypothesis in the model includes appearance, location and scale components with relative weights λ i /(λ 1 + λ 2 + λ 3 ), i = 1, 2, 3, learnt automatically by the algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Because given a set S i , we can only observe the jumping time of the events in each counting process without knowing the identity of the covered items (which is a key difference from [8]), the best thing we can do is to sample from these historical data.
Sentence 2: let the number of counting processes that a single item s ∈ V is involved to induce be N s , and the collection of all the jumping timestamps before time T be J s . Then, for the s-th entry of τ , with probability |J s |/nN s , we uniformly draw a sample from J s ; and with probability 1 − |J s |/nN s , we assign a time much greater than T to indicate that the item will never be covered until infinity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: guarantee like where R is increasing in each kg t k. For example, when the convex space W has diameter D, As foreshadowed in Section 3, we will need to consider the case of biased gradients.
Sentence 2: e[g t ] = rF (w t ) + b t for some unknown bias vector b t . Given these biased gradients, a natural question is: to what extent does controlling the regret R T (w ? )
Relationship: entailment
Domain: NeurIPS
Sentence 1: We now compare our algorithm against other dimensionality reduction techniques.
Sentence 2: in this paragraph we present head-to-head comparisons for the following five methods: (i) SVD: the Singular Value Decomposition (or Principal Components Analysis) dimensionality reduction approach -we use MatLab's svds function; (ii) LLE: the famous Local Linear Embedding algorithm of [18] -we use the MatLab code from [23] with the parameter K determining the number of neighbors setting equal to 40; (iii) LS: the Laplacian score feature selection method of [10] -we use the MatLab code from [22] with the default parameters 2 ; (v) HD: we run the k-means algorithm on the High Dimensional data; and (vi) RP: the random projection method we proposed in this work -we use our own MatLab implementation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: R3) At the points {z} where a * i z = 0 for certain i ∈ M, the corresponding weight will be w i = 0.
Sentence 2: the losses (z; ψ i ) in (2) that are nonsmooth at points z will be eliminated, to prevent their contribution to the reweighted gradient update in (10).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Although these update equations do require that we differentiate with respect to a function of another gradient term, such "double backprop" terms (see e.g., Drucker and Le Cun [1992]) are easily computed by modern automatic differentiation tools.
Sentence 2: we propose the regularized update (4) The intuition of this regularizer is perhaps most easily understood by considering how it changes the Jacobian at equilibrium (though there are other means of motivating the update as well, discussed further in Appendix F.2).
Relationship: entailment
Domain: NeurIPS
Sentence 1: It follows that with a formulation like (3), the only way to utilize unlabeled data is to replace K by a kernel K in (3), or k by k in (2), where K (or k) depends on the unlabeled data.
Sentence 2: the only benefit of unlabeled data in this setting is to construct a good kernel based on unlabeled data.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Supposing D XY satisfies Tsybakov noise, we define a quantity based on a standard generalization bound for passive learning [12].
Sentence 2: [12] have shown that, for any V ⊆ C, with probability at least 1 − δ/(4km 2 ), sup h,g∈V Consider the following modification of k-batch CAL, designed to be robust to Tsybakov noise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The coupling between the systems will be linear as described by the connectivity matrix W. Using this factorization, approximate state estimation can then be efficiently performed via approximations of loopy belief propagation (BP) [14].
Sentence 2: we show that the loopy BP updates at each of the factor nodes associated with the integrate-and-fire and calcium imaging can be performed via a scalar standard forward-backward filter.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We incorporate a target density ρ n (or simply ρ, when n is clear from the context) by normalizing W so that W = 1 and taking G to be a sample from G n (ρW ).
Sentence 2: we set Q = H n (ρW ) = ρH n (W ) and then connect i to j with probability Q ij , independently for all i < j. Stochastic block models are specific examples of W -random graph in which W is constant on sets of the form I i × I j , where (I 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let p(y) be a two-component mixture of a (univariate) Bernoulli distribution.
Sentence 2: let φ(x) and ψ(z) be the identity maps, Λ provide the mean of the Bernoulli, and p(X = 0.4) = 1/2, p(X = 0.6) = 1/2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Q-learning with locally consistent data: An alternative approach is to simply maintain a single regressor, but ensure that any batch of Q-labels is self-consistent before updating the regressor.
Sentence 2: given a batch of training data and the current regressor, we first create a single set of consistent labels for each example (see below), then update the regressor using these labels.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let us observe that if we have a minibatch and normalize a particular node x using either the minibatch statistics or their moving averages, then the results of these two normalizations are related by an affine transform.
Sentence 2: let µ be an estimate of the mean of x, and σ be an estimate of its  standard deviation, computed perhaps as a moving average over the last several minibatches.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Consider the pairwise binary graphical model for the independent set problem on G = (V, E) given by the edge potentials ψ ij (x i , x j ) = 1 − x i x j for all (i, j) ∈ E. We can easily check that the 6-cycle has 18 distinct independent sets while the disconnected cover has only 16 independent sets.
Sentence 2: the disconnected 2-cover does not provide an upper bound on the number of independent sets in all 2-covers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: This paper showed that the SAG iterations achieve a much faster convergence rate, but the SAG iterations may also be advantageous in terms of tuning step sizes and designing termination criteria.
Sentence 2: the SAG iterations suggest a natural termination criterion; since the average of the y k i variables converges to g (x k ) as x k −x k−1 converges to zero, we can use (1/n) i y k i as an approximation of the optimality of x k . Further, while SG methods require specifying a sequence of step sizes and mispecifying this sequence can have a disastrous effect on the convergence rate [7, §2.1], our theory shows that the SAG iterations iterations achieve a linear convergence rate for any sufficiently small constant step size and our experiments indicate that a simple line-search gives strong performance.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Define Q ::@, p-1 + C T R- 1 C, and M ::@, Q + K. Suppose the cutting matrix K is symmetric and positive semidefinite.
Sentence 2: then we are guaranteed that we have the bounds: :::; p(M-1 K) :::; Amax (K) (4) It should be noted that the conditions of this theorem are sufficient, but by no means necessary, to guarantee convergence of the ET algorithm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: (FollowTheLeader) Perhaps the most natural algorithm one might think of is to simply select x t as the best point in hindsight.
Sentence 2: the learner can choose (BeTheLeader) When the learner is prescient, then we can do slightly better than FollowTheLeader by incorporating the current loss function: x t = arg min x∈K t s=1 s (x).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Tractable flows-based model Parameterizing the variational distributions with flows is proposed to mitigate the limitation of expressive ability of the variational distribution.
Sentence 2: assuming a series of invertible transformations as {T t : R r → R r } T t=1 and z 0 ∼ q 0 (z|x), we have −1 by the change of variable formula.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As described in [30], the mIBP defines a distribution over an unbounded set of binary Markov chains, where each chain can be associated to a binary latent feature that evolves over time according to Markov dynamics.
Sentence 2: if we denote by M the total number of feature chains and by T the number of observable components (event mentions), the mIBP defines a probability distribution over a binary matrix F with T rows, which correspond to observations, and an unbounded number of columns (M → ∞), which correspond to features.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let A be a learning algorithm which, based on the sample S, constructs a hypothesis (estimator) h from some set of hypotheses H. Denoting by (y, h(x)) the instantaneous loss of the hypothesis h, we wish to assess the true loss L(h) = E µ (y, h(x)) where the expectation is taken with respect to µ.
Sentence 2: the objective is to provide data-dependent bounds of the following form.
Relationship: entailment
Domain: NeurIPS
Sentence 1: There are 17, 770 and 624, 961 items in netflix and yahoo-music, respectively.
Sentence 2: we obtain the user embeddings {w i } 2 R k   and item embeddings h j 2 R k by the standard low-rank matrix factorization [4] with k 2 {50, 200}.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For non-parametric regression with the square loss, one pass SGM has been studied in, e.g., [29,22,26,9].
Sentence 2: [29] proved capacity independent rate of order O(m − 2ζ 2ζ+1 log m) with a fixed step-size η m − 2ζ 2ζ+1 , and [9] derived capacity dependent error bounds of order O(m − 2 min(ζ,1) 2 min(ζ,1)+γ ) (when 2ζ + γ > 1) for the average.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Oracle Problem: We develop our candidate solution ( L, Ĉ) by considering the alternate optimization problem where we add constraints to (2) based on what we hope its optimum should be.
Sentence 2: recall the SVD of the true L 0 = U 0 Σ 0 V ⊤ 0 and define for any matrix X the projection onto the space of all matrices with column space contained in U 0 as P U0 (X) := U 0 U ⊤ 0 X.
Relationship: entailment
Domain: NeurIPS
Sentence 1: 2b: Bounding p t ∞,u from a warm start Unfortunately, we cannot apply our results of step 2a directly since we are only assuming that X 0 has a warm start, not a stationary start.
Sentence 2: we only assume that X 0 − Ỹ0 2 < ω for some ω > 0, where Ỹ0 ∼ π is at the stationary distribution.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Best of the three initialization methods, by starting at the hybrid algorithm results, the search algorithm in very few steps can find  the local optimum, which often happened to be the global optimum as well.
Sentence 2: it only takes 1 or 2 steps of search in the 10-node chain case and 1 to 3 steps in the 50-node tree case.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The AP loss of an output R is defined as 1 − AP(R * , R), where AP(•, •) corresponds to the AP of the ranking R with respect to the true ranking R * .
Sentence 2: the parameters are obtained by solving the following convex optimization problem: The computational complexity of solving the above problem depends on the complexity of the corresponding loss-augmented inference, that is, For a given set of parameters w, the above problem requires us to find the most violated ranking, that is, the ranking that maximizes the sum of the score and the AP loss.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Lemma 1 links the random variables W n and W n to the Hausdorff distance and the coverage for R and R n . Thus, we call them coverage random variables.
Sentence 2: now we define the L 1 and L 2 coverage risk for estimating R by R n as Risk 1,n (and Risk 2,n ) is the expected (square) projected distance between R and R n . Note that the expectation in (4) applies to both R n and U R . One can view Risk 2,n as a generalized mean integrated square errors (MISE) for sets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: However, there is a unique, so-called adapted system of modules, for which this issue does not arise.
Sentence 2: if, as is usually done, we let the indexing set T m be the set of Standard Young Tableaux (SYT) of size m (see Appendix A in the supplementary materials for the exact definition), such as then the adapted modules at different levels of the coset tree are connected via where is the tableau that we get by removing the boxes containing n − k + 1, . . . , n from t . We also extend these relationships to sets in the obvious way: We will give an explicit description of the adapted modules in Section 4.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, with probability 0.99, q T − u 2 ≤ ǫ, where q T is the T -th iterate of Algorithm 1.
Sentence 2: algorithm 1 obtains an ǫ-accurate solution with number of samples (n) given by: ǫ 2 log((σ 2 + .75)/(σ 2 + .5)) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can also express the input constraint l j ≤ x j ≤ u j as a quadratic constraint, which will be useful later.
Sentence 2: l j ≤ x j ≤ u j if and only if (x j −l j )(x j −u j ) ≤ 0, thereby yielding the quadratic constraint x 2 j ≤ (l j +u j )x j −l j u j .
Relationship: entailment
Domain: NeurIPS
Sentence 1: We refer to the set of edges that share the ith parameter p i by E i = {e ∈ E|π(e) = i}.
Sentence 2: for any e 1 , e 2 ∈ E i , π(e 1 ) = π(e 2 ) and hence w e1 = w e2 = p π(e1) .
Relationship: entailment
Domain: NeurIPS
Sentence 1: When it comes to networks, triangular motifs (Figure 1) are already of significant interest in biology [13], social science [19,9,10,16], and data mining [21,18,8].
Sentence 2: 2-and 3-edge triangular motifs are central to the notion of transitivity in the social sciences -if we observe edges A-B and B-C, does A have an edge to C as well?
Relationship: entailment
Domain: NeurIPS
Sentence 1: Though training DNNs with softmax or SVM as output activation is effective in many tasks, it is possible that alternative activations that consider manifold structure of data by interpolating the output based on both training and testing data can boost performance of the network.
Sentence 2: resNets can be reformulated as solving control problems of a class of transport equations in the continuum limit [21,5].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Assuming Gaussian sensing vectors however, such nonconvex paradigms can provably locate the global optimum, several of which also achieve optimal (statistical) guarantees.
Sentence 2: starting with a judiciously designed initial guess, successive improvement is effected by means of a sequence of (truncated) (generalized) gradient-type iterations given by where z t denotes the estimate returned by the algorithm at the t-th iteration, µ t > 0 is learning rate that can be pre-selected or found via e.g., the backtracking line search strategy, and ∇ (z t , ψ i /y i ) represents the (generalized) gradient of the modulus-or squared modulus-based LS loss evaluated at z t . Here, T t+1 denotes some time-varying index set signifying the per-iteration gradient truncation.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Define S := B(x, r) ∩ M for a point x ∈ M . Then where r 1 = τ − τ 1 − 2r/τ .
Sentence 2: if r ≤ ǫτ /72d for 0 ≤ ǫ < 1, then Lemma 8 (Separation) Assume that we pick k, r and R to satisfy the conditions: Then with probability 1 − δ, we have: (1) All points in A σ−r and A ′ σ−r are kept, and all points in S σ−r are removed.
Relationship: entailment