sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: The RGS (m,k) algorithm is useful in situations where the amount of time to produce an answer may be limited and/or the limit unknown beforehand: it is easy to show that the growing phase of GS (m) produces an an upper-bound of the m-boundary of X.
Sentence 2: the RGS (m,k) algorithm, if interrupted, will return an approximation of this upper bound.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, when there are d outputs, the rank of L is d and the method trains ridge regression d times, but the kernel PCA step first decorrelates the outputs.
Sentence 2: in the special case of multiple outputs regression with a linear kernel , the method is also related to the work of [2] (see [4, page 73] for an overview of other multiple output regression methods.)
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To obtain a risk bound for G i ab , we need to find a closed-form expression for R S (G i ab ).
Sentence 2: for this task, let U [a, b] denote the uniform distribution over [a, b] and let σ i a,b (x) be the probability that a ball with center x i assigns to x the class label y i when its radius ρ is drawn according to U [a, b]: Consequently, the risk R (x,y) (G i ab ) on a single example (x, y) is given by G i ab (x) if y = 0 and by 1 From this expression we see that R S (G i ab ) is small when G i ab (x j ) → y j ∀j ∈ i.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Also, X y(x) represents the set of samples that have labels equal to the label of x and X ¬y(x) the data that have labels opposite to the label of x.
Sentence 2: the estimator for the entropy of the data that is classified with projection π is as follows: From 3 and using the fact that = π] for which we use the notation we estimate the objective as the contribution of each data point to the objective corresponds to a distance ratio on the projection π * where the class of the point is obtained with the highest confidence (data is separable in the neighborhood of the point).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Being parallelizable (d = 1, 2, 4 or 8), manifold of unit quaternions enjoys an injective homomorphism to the orthonormal matrix ring composed of the orthonormal basis [35].
Sentence 2: we define V : S 3 → R 4×4 as follows: It is easy to verify that V(q) is orthonormal for every q ∈ S 3 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: An algorithm is called p,-margin successful if, for any input labeled sample, it outputs a hypothesis half-space that classifies correctly as many sample points as any half-space can classify correctly with margin p, (that is, discounting points that are too close to the separating hyper-plane).
Sentence 2: a p,-margin successful algorithm is required to output a hypothesis with close-to-optimal performance on the input data (optimal in terms of the number of correctly classified sample points), whenever this input sample has an optimal separating hyper-plane that achieves larger-than-p, margins for most of the points it classifies correctly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We establish that K(δ; p, d, r) is O p 2η+2 d 2η r 5 δ −1 poly log(p, d, r, δ −1 ) in [15].
Sentence 2: we require the number of samples in (9) scaling as n = Ω p 4η+4 d 4η r 10 δ −2 −2 poly log(p, d, r, δ −1 ) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We further use the Cholesky decomposition to parametrize the shared covariance structure across voxels: U = LL T , where L is a lower triangular matrix.
Sentence 2: β i can be written as where α i ∼ N (0, I) (this change of parameter allows for estimating U of less than full rank by setting L as lower-triangular matrix with a few rightmost-columns truncated).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (Note that N ij  k has been implied by N ⌫ 0 and ) Since dropping the rank constraint eliminates the constraints B 2 Im(N ) and D 2 Im(N ) in ( 16) when N 0 [32], we obtain the following relaxed problem, which is jointly convex in N , B and D: Unfortunately, nonlinear semidefinite optimization problems in the form (20) are generally thought to be too expensive in practice despite their polynomial theoretical complexity [33,34].
Sentence 2: we develop an effective training algorithm that exploits problem structure to bypass the main computational bottlenecks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The previous result generalizes to a sigmoidal unit O = σ(S) = 1/(1 + ce −λS ) trained to minimize the relative entropy error The terms O and I i are not independent but using a Taylor expansion with the N W GM approximation gives with S EN S = j w j p j I j .
Sentence 2: as in the linear case, the expectation of the dropout gradient is approximately the gradient of the ensemble network regularized by weight decay terms with the proper adaptive coefficients.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , M T . To ensure this balance, we use the regularization term stated below: The trade-off parameter γ t controls the regularization of M t for all t = 0, . . . , T . If γ 0 → ∞, the shared metric M 0 reduces to the plain Euclidean metric and if γ t>0 → ∞, the task-specific metrics M t>0 become irrelevant zero matrices.
Sentence 2: if γ t>0 → ∞ and γ 0 is small, we learn a single metric M 0 across all tasks.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, a REVERSE-PUSH(v, k) operation involves updating the residuals for d in (v) + 1 states.
Sentence 2: note that t∈S p k e v [t] = 1 and hence, via a straightforward counting argument, we have that for any v ∈ S, t∈S 1 {p k e v [t]≥δr} ≤ 1/δ r . we have: Finally, we choose δ r = 2 δ max log( max/pf ) to balance T f and T r and get the result.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We also find that different with text classification and sequential labelling, both ρ > 1 and ρ < 1 can obtain optimal scores.
Sentence 2: with the feature configuration of MSTParser, the value of w ⊺ Φ(x, y * ) may not always be positive during the online learning process, and it reflect the fact that feature space of parsing problems is usually more complex.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We discern from the graph that definition 2 is satisfied because: ( 1 2) P (X|R y , Z) = P (X|R y ) and R y is the minimal set such that X⊥ ⊥({R y , Z} \ R y )|R y and (3) P (Z|R y ) = P (Z) and ∅ is the minimal set such that Z⊥ ⊥R y |∅.
Sentence 2: the order Y < X < Z < R y induces a general ordered factorization P (X, Y, Z, R y ) = P (Y |X, Z)P (X|R y )P (Z)P (R y ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Instead of using η m to localize the training error of data points, we localize a model's contribution ŷm = ξ(x) T w m towards the global fit of training point y, similar to how LWR operates during test time (Eq.4).
Sentence 2: already during training, local models must collaborate to fit a data point ŷ = ∑ m=1 η m (x)ξ(x) T w m . Our loss function is combining the localizer η m (x n ) and the This form of localization couples all local models, as in classical radial basis function networks [19].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, there exists i 0 such that if i > i 0 then |l(y, λ * i , x) − l(y, λ * ∞ , x)| < for any y ∈ Y and x ∈ X .
Sentence 2: input: Countable set of experts {H k,h }, y 0 ∈ Y, λ 0 ∈ Λ, initial probability {α k,h }, For n = 0 to ∞ Play y n , λ n . Nature reveals x n Suffer loss l(y n , λ n , x n ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ICA, Product-of-Experts and Fields-of-Experts), can be interpreted as a form of feed-forward neural network.
Sentence 2: the terms {∂E/∂x i } and {∂ 2 E/(∂x i ) 2 } can be efficiently computed using a forward and backward pass: the first pass performs forward inference (computation of E(x; w)) and the second pass applies the backpropagation algorithm [3] to obtain the derivatives of the energy w.r.t.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In defining ∆ Ŝi = ( Ŝi − S i ) we find that The fluctuations of the sufficient statistics ∆ Ŝi is a random variable with mean zero, and the randomness of (g L i − g i ) enters only via λ i .
Sentence 2: one can assume a very small statistical correlation between those two terms, the cross-term can be expected to be negligible.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This can be seen as a semiparametric statistical model [22,23], which includes not only parameters of interest, but also additional nuisance parameters with possibly infinite DoF, where only the policy is modeled parametrically and the other unspecified part corresponds to the nuisance parameters.
Sentence 2: model-free WLPS can be framed as a semiparametric statistical inference problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We want to show that given a sequence of matrices §¦ X £ £ s s £ ' £ £ ¡ ' £ s £ £ ¡ £ £ ¤ $ £ s I) .
Sentence 2: ¥ ¥ AH ¤¡ ¦ X ¤ $ £ ¡ ) £ ¤ $ £ s H) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To address the above-mentioned issue, we extend the Rademacher random variables to the complex domain and consider all roots of z m = 1, that is, Ω = {ω j } m−1 j=0 where ω j = e i 2πj m . Suppose σ(i) is a Rademacher random variable with Pr[σ(i) = ω i ] = 1/m.
Sentence 2: by elementary algebra, E[σ(i) p ] = 0 whenever m is relative prime to p or m can be divided by p. by setting m = 4 we avoid collisions of repeated entries in a 3rd order tensor.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As a result, we know t k ≥ t min := min i=0,...,k t i ≥ η/β.
Sentence 2: it is easy to see that in the k-th iteration of Algorithm 4, x * is contained in two balls: we have the following convergence result for Algorithm 4, whose proof is similar to that for Algorithm 3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using the same derivation as in Remark 2, the probability of error ℓ(n) for this strategy may be bounded as In the Uniform+UCB-E allocation strategy, i.e., a two-level algorithm that first selects a bandit uniformly and then pulls arms within each bandit using UCB-E, the total number of pulls for each bandit m is k T mk (n) = n/M , while the number of pulls T mk (n) over the arms in bandit m is determined by UCB-E.
Sentence 2: the probability of error of this strategy may be bounded as where the first inequality follows from Theorem 1 in [1] (recall that H m = k b 2 /∆ 2 mk ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The geometrical interpretation is that q i,j requests whether the reference r σ is closer to object θ i or object θ j in R d . Consider the line connecting θ i and θ j in R d . The hyperplane that bisects this line and is orthogonal to it defines two halfspaces: one containing points closer to θ i and the other the points closer to θ j .
Sentence 2: q i,j is a membership query about which halfspace r σ is in, and there is an equivalence between each query, each pair of objects, and the corresponding bisecting hyperplane.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, at high dimensions, the relative error made as considering E||X|| instead of the random variable ||X|| becomes asymptotically negligible [2].
Sentence 2: the following holds in high dimensions: Finally, E(max(W )) ≤ √ 2σ ln(N ) when W follows a Gaussian distribution [13], establishing the following: Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Depending on the value of b, we will have n + 1 possible sums, respectively − k i=1 c r(i) + n i=k+1 c r(i) , k = 0, . . . , n. It is obvious that those sums only depend on the order of the products u • x i , i = 1, . . . , n. When u varies smoothly on the unit circle, as the dot product is a continuous function of its arguments, the changes in the order of the dot products will occur only when there is a pair (i, j) such that u • x i = u • x j .
Sentence 2: there are at most as many order changes as there are pairs of different points, i.e., n(n − 1)/2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Notice that e) where e is the identity permutation, a quantity which is also called the number of inversions of σ.
Sentence 2: in contrast to the previous mappings, the norm φ L (σ) is not constant for any σ ∈ S K . Hence it is not possible to write the loss ∆ L (σ, σ ) as − φ L (σ), φ L (σ ) 2 .Moreover, this mapping is not distance preserving and it can be proven that 1 Wang et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consequently, a method for approximating ∂J(θ) would consist in estimating I h for a sufficiently small h. We know that J(θ) can be numerically estimated by 1 M M m=1 J N ωm (θ).
Sentence 2: it seems natural to estimate I h by where we used independent random numbers to evaluate J(θ + h) and J(θ − h).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this case, we are able to show that if we allow the low rank matrix B output to be of rank k + 4, then one can achieve the relative error guarantee of (1.2) with respect to the best rank-k approximation, namely, that F . Further, our algorithm runs in a sublinear n • k • poly(1/ ) amount of time.
Sentence 2: our lower bound ruling out sublinear time algorithms achieving (1.2) for bicriteria algorithms cannot hold for this class of matrices.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, the similarity graph built by our method has a locality preserving property by the definition of the weights.
Sentence 2: we can use the adjacency matrix, W [i], of the i-th cluster as a similarity between points in the corresponding manifold and obtain a low-dimensional embedding of the data by taking the last few eigenvectors of the normalized Laplacian matrix associated to W [i] [3].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because J is strongly-connected, the Perron-Frobenius theorem guarantees a simple eigenvalue equal to ρ(J) and a corresponding all-positive eigenvector.
Sentence 2: when β crosses 1/ρ(J) from below, the Perron-Frobenius eigenvalue of f (m) crosses 1 from below, giving rise to a supercritical pitchfork bifurcation with two new stable steady-states corresponding to the Perron-Frobenius eigenvector.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unlike many existing parametric survival models, our model neither assumes a parametric form for the interactions between the covariates and the survival times, nor does it restrict the distribution of the survival times to a parametric model.
Sentence 2: it can flexibly describe non-proportional hazard rates with complex interactions between covariates and survival times, which are common in many diseases with heterogeneous phenotypes (such as cardiovascular diseases [2]).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Similarly, if one were to assume that a simple operation (such as computing a probability vector over categories using Bayes rule) would take on the order of £ § sec per word, then a full pass through £ ¨ § © words will take 1000 seconds.
Sentence 2: algorithms that make multiple passes over this sized corpus (such as occurs in many clustering and classification algorithms) will have run times in days.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We consider a positive vector-valued step-size ∈ ℝ + and "•" denotes component-wise (Hadamard) product of two vectors.
Sentence 2: again, by exploiting (4), since where is an estimated eigenvalue of J as given in (2), when H is a symmetric matrix, its eigenvalue is given by eig we can update the step size component-wise by Since the mapping ℳ in SGD involves the gradient g(w ( ) ; B ( ) ) of a randomly selected training example B ( ) , ℳ is itself a random variable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This result improves upon [9] where it is proven only for a global minimizer.
Sentence 2: proof: If g is any function in H such that g(x j ) = 0, j ∈ IN m and t a real number such that |t| g ≤ , for > 0, then we have that f 0 2 ≤ f 0 + tg 2 from which it follows that (f 0 , g) = 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We call a weighted data set C a (k, ε)-coreset for another (possibly weighted) set  Hereby φ(C | θ) is generalized to weighted data sets C in the natural way (weighing the contribution of each summand x j ∈ C by γ j ).
Sentence 2: as ε → 0, for a sequence of (k, ε)-coresets C ε we have that Further, under the additional condition that all variances are sufficiently large (formally λ∈spec(Σi) λ ≥ 1 (2π) d for all components i), the log-normalizer ln Z(θ) is negative, and consequently the coreset in fact provides a multiplicative (1 + ε) approximation to the log-likelihood, i.e., ( More details can be found in the supplemental material.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We also introduce skip connections from the encoder to the decoder in the CL model, where each intermediate blocks' outputs are upsampled and concatenated with o t as well, so the decoder can take advantage of the localized memory provided.
Sentence 2: with k blocks, we use a stride of 2 in each block, so the output of the k-th block must be upsampled by a factor of 2 k . only in the CL model do we also concatenate the the intermediate LSTM cell outputs to the input to the decoder D. These skip-connections from the intermediate blocks are a way to propagate spatio-temporal memory to the decoder at different spatial scales, taking advantage of the specific structure of this problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The MAP problem is then equivalent to the following linear program: where The extreme points of the marginal polytope are integral and correspond one-to-one with assignments x.
Sentence 2: there always exists a maximizing µ that is integral and corresponds to x M . Although the number of variables in this LP is only O(|E| + |V |), the difficulty comes from an exponential number of linear inequalities typically required to describe the marginal polytope M(G).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, such a bound is applicable only to the bounded-support class P B , and helps to remove the dependence on √ d but at the cost of sacrificing the rate (i.e., from m −1/2 + n −1/2 to m −1/4 + n −1/4 ).
Sentence 2: such an upper bound matches the lower bound in Corollary 2 for ReLU networks over the network parameters, but not in terms of the sample size, and is interesting only in the regime when d max{n, m}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Y t where Y If the matrix that we would like to learn is asymmetric, we can transform it to an equivalent symmetric model by defining a new data matrix Y = 0 Y Y T 0 .
Sentence 2: for the remainder of the paper, we will assume a n × n symmetric matrix which takes values in [0, 1] (real-valued or binary), but as argued above, our results apply more broadly to categorical-valued asymmetric matrices.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When changing the means of arms below, the triggering probability of arms above is not changed.
Sentence 2: if we change µ to µ backwards, the triggering probability of each arm is unaffected before its expectation is changed, and when changing the mean of an arm i, the expected reward of the action is at most changed by p D,S i The proof of Lemma 2 is more complex, since the bottom-up modification does not work directly on graphs with cycles.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If the algorithm converges, i.e., the sequence (ŝ e,t ) converges to a point s * , and f i (•) is continuous, the variance of sEM-vr will converge to zero, while that of sEM will remain positive.
Sentence 2: sEM-vr has asymptotically smaller variance than sEM, and we will see that this leads to better asymptotic convergence rates.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From Lemma 1 we deduce that if the per-tree budget is at least we obtain 1−γ ≤ for any policy π. ASOP aggregates the trees and computes the optimal policy π of the resulting empirical MDP whose transition probabilities are defined by the frequencies (over the m S3-trees) of transitions from state-action to successor states.
Sentence 2: π is actually a policy maximizing the function If the number m of S3-trees and the per-tree budget n are large, we therefore expect the optimal policy π of the empirical MDP to be close to the optimal policy π * of the true MDP.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For each state s, we define a continuous observation probability distribution, for observing a pdf Pt(x) in state s. Next, the initial state distribution {1fsLES of the HMM is given by the uniform distribution, 1fs = liN, with N = lSI being the number of states.
Sentence 2: each state is a-priori equally probable.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Parameter choices: As with previous GP-based algorithms that use confidence bounds, our theoretical choice of (i) in TRUVAR is typically overly conservative.
Sentence 2: instead of using ( 14) directly, we use a more aggressive variant with similar dependence on the domain size and time: , where t (i) is the time at which the epoch starts, and a is a constant.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The event pixel address was used to produce a 40-dimensional embedding via a learned embedding matrix [9], and combined with the polarity to produce the input.
Sentence 2: the network architecture was 41-110-10 for the Phased LSTM and 42-110-10 for the BN-LSTM, with the time given as an extra input dimension to the BN-LSTM.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By the classic analysis of greedy algorithm for submodular maximization, we can conclude that every item we add has an added value that is at least (1 − )(L − f (S))/|OPT|.
Sentence 2: after adding |OPT| ln(L)/(1 − ) items, f (S) becomes at least L. To upper bound rounds, we divide the rounds into two groups.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This invariance follows from the decomposition of equation ( 1): in particular, the distribution pi(n) (x; Tn) is left invariant by reparameterization; and TRP does not change terms in ri(n) (x; Tn).
Sentence 2: the overall distribution remains invariant -i.e., p(x; Tn) == p(x; TO) for all n. By continuity of the map T f-7 p(x; T) , it follows that any fixed point T* of the algorithm also satisfies p(x; T*) == p(x; TO).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since e where e is a vector of 1's, it is straightforward to show that the following inequality In addition, the equality holds if and only if Therefore, |V 1 | β |V 2 | β achieves the maximum value when two clusters are of equal weighted size.
Sentence 2: minimizing SRcut is equivalent to minimizing the similarity between two clusters and, at the same time, searching for a balanced partition.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If G : [0, ∞) → R denotes the cumulative distribution function of the squared deviation of F (p 1 , .
Sentence 2: . . , pk ) from its mean, then We then have a mean squared error of It should be noted that the constants in both the bias bound and the variance bound depend exponentially on the dimension d. Lower bounds in terms of d are unknown for estimating most density functionals of interest, and an important open problem is whether this dependence can be made asymptotically better than exponential.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Because of memory limitations, [25] actually solved an approximate version of the PE objective function, where rather than using all N 2 pairwise point interactions, only BN interactions are used, corresponding to using minibatches of B = 5 000 points.
Sentence 2: the solution obtained is not a minimizer of the PE objective, as can be seen from the higher objective value in fig.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The hyperparameter 0 controls the impact of the GAN objective functions.
Sentence 2: we use a VAE-like objective function to model the cycle-consistency constraint, which is given by where the negative log-likelihood objective term ensures a twice translated image resembles the input one and the KL terms penalize the latent codes deviating from the prior distribution in the cycle-reconstruction stream (there are two KL terms).
Relationship: reasoning