sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Note that care must be taken to ensure that the addition of this penalty term does not violate the nonnegativity of c, and to ensure that the argument of ψ −1 lies in its domain.
Sentence 2: given a = 1, with appropriate penalty functions, our solution to (3.6) can be utilized for maximizing entropy of Bc subject to linear or non-linear constraints on c. , where h is a "link" function that models a nonlinear relationship between A and the approximant BC, we may wish to minimize D ϕ (h(BC), A).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Experimental results demonstrate their superior performance.
Sentence 2: let Δ = ρ 1 − ρ 2 ∈ (0, 1], and μ :=   and the number of calls to the least squares solver of In order to satisfy the CCA constraints, we perform a last normalization And we output (û, v) as our final approximate solution to (1).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The product of an arbitrary vector v by the GGN matrix, Gv = J N H L J N v, amounts to the sequential multiplication of v by three matrices.
Sentence 2: ( 7) and setting l = z, p(z|a) can be re-written as where S is the set of every possible label sequence and f z = log π∈B −1 (z) exp(b π ) is the logsum-exp function 1 , which is proportional to the probability of observing the label sequence z among all the other label sequences.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Hence, for a measurable function f : where we used the fact that i+1 (x) ≤ 1 and η is an appropriate multiple of η, e.g.
Sentence 2: for x, y ∈ K, ρ(x, y) is the distance in the Riemannian metric whose metric tensor is the Hessian of F . the metric tensor on the tangent space at x assigns to a vector v the length and to a pair of vectors v, w, the inner product v, w x := D 2 F (x)[v, w].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Furthermore, though the kernel learning algorithm was not expoed to the test set noise, the learned kernel reflects better the structure of the feature space which makes the learned kernel more robust to noise.
Sentence 2: 5 is a scatter plot comparing the test error of each of the binary classifiers when trained with the original RBF a kernel versus the performance achieved on the same binary problem with a learned kernel.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Intuitively, if during training an algorithm finds a box with large volume, such that all weight-vectors belonging to it attain low training error and are simple, we expect the classifier based on the center of mass to be robust to noise or fluctuations.
Sentence 2: if w attains the supremum sup w∈Q R(w) then so does at least one vertex.
Relationship: neutral
Domain: NeurIPS
Sentence 1: While one advantage of Equation 2is that all connections are assigned a non-zero score, another benefit is derivatives with respect to model parameters are easy to determine and compute using the standard formula for the derivative of an inverse matrix.
Sentence 2: given the sensitivity of model performance on signal processing illustrated in Figure 2, our goal is now to learn the optimal filter α by optimizing a smooth loss function.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To see why, recall that quantities max i∈[m] ψ i and (1/m) m i=1 ψ i have magnitudes on the order of √ m x and π/2 x , respectively, whereas h ≤ ρ x for some small constant 0 < ρ ≤ 1/10, to be discussed shortly.
Sentence 2: tGGF requires at most O(log(1/ )) iterations to achieve any given solution accuracy > 0 (a.k.a., dist(z t , x) ≤ x ), with iteration cost O(mn).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Combining these aspects together, the R-MSN is able to help clinicians evaluate the projected outcome of a complex treatment scenario -providing timely clinical decision support and helping them customize a treatment regimen to the patient.
Sentence 2: encoder The goal of the encoder is to learn good representations for the patient's current clinical state, and we do so with a standard LSTM that makes one-step-ahead predictions of the outcome ( Ŷt+1 ) given observations of covariates and actual treatments.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Discretizing logistic loss f (x) = log(1 + e −w T x ) does not fit exactly into the setting of Theorem 1 due to nonexistence of x * .
Sentence 2: we introduce a second-order ODE that generates an accelerated first-order method for smooth functions if we simply discretize it using any Runge-Kutta numerical integrator and choose a suitable step size.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then, if x is the closest sample point to x , we have for n sufficiently large, |x − x| ≤ τ 0 and f (x) ≥ f (x 0 ) + δ/2 and thus f , contradicting the fact that x is the maximizer of the KDE over samples in its τ -radius neighborhood.
Sentence 2: then x 1 and x 2 are points which respectively attain the highest values of f on P 1 and P 2 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Poisson sampling is executed in the following way: proceed down the list of elements and carry out one randomized experiment for each element, which results either in the election or in the nonselection of the element [16].
Sentence 2: theoretically, we provide the error bound analysis, which supplies a justification for the algorithm and give a tradeoff between the subsample size and approximation efficiency.
Relationship: neutral
Domain: NeurIPS
Sentence 1: From this equation, we may infer the following: If P t R t = k, then the FDR will be bounded by ↵ as long as the total alpha-wealth P t ↵ t that was used for testing is smaller than k↵.
Sentence 2: instead of presenting the most general and improved algorithms immediately, we choose to present results in a bottom-up fashion, introducing one new concept at a time so as to lighten the symbolic load on the reader.
Relationship: neutral
Domain: NeurIPS
Sentence 1: One natural next step should be to generate direct quantitative predictions of behavior in one task based on a model trained on another task -ideally on an individual subject level, and in a task that fits in our framework that has not been extensively explored (for example, an internal-context Flanker variant, or a context-dependent response congruence judgment task).
Sentence 2: figure 2 shows mean RTs and accuracies produced by our model for the AX-CPT, under the same parameters that we used for the flanker model.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, each corrupted row/column item has perturbed feature x i = u i + ∆u i and y i = v i + ∆v i , where ∥∆u∥ ∞ ≤ ξ 1 and Theorem 3 suggests that, if features have good quality in the sense that items with corrupted features are not too many, for example g(n), h(n) = O(log n), then sample complexity of DirtyIMC can be O(n log n √ log n) = o(n 3/2 ) as well.
Sentence 2: we propose a dirty statistical model which balances between feature and observation information simultaneously to complete a matrix.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To provide insight into the nature of our algorithms, we focused on a single non-trivial domain.
Sentence 2: the motivation for this approach comes from the recursive nature of tree search: if the heuristic can be adjusted to match the value of a deep search of depth D, then a search of depth k with the new heuristic would be equivalent to a search of depth k + D with the old heuristic.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, we show that this algorithm needs to query only O(k log n) such e I .
Sentence 2: let P : [n] → R be a distribution, and let Q be the empirical distribution after O( k+log 1/δ ε 2 ) samples from P . Then with probability at least 1 − δ, we have that P − Q A k ≤ ε.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In contrast, methods such as DR, AP and the primal-dual hybrid gradient descent (PDHG) proposed in [29] used for SSL on hypergraphs [17], require performing a complete gradient descent and computing a total of R projections at each iteration.
Sentence 2: suppose that W = βD, where β is a hyper-parameter, and D is a diagonal degree matrix such that Dii Djj log 1 ) iterations to return an -optimal solution.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Probabilistic classification performance is measured by the conditional logloss (the negative conditional likelihood), logloss Ptrg(X of the estimator, P (Y |X), under an evaluation distribution (i.e., the target distribution, P trg (X)P (Y |X), for the sample selection bias setting).
Sentence 2: expert-provided source distributions, feature functions, and constraint statistic values, respectfully denoted P � src (x), f � (x, y), and c � , can be specified to express a range of assumptions about the conditional label distribution and how it generalizes.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For example, we may assume ε i ∼ N (0, σ 2 I D ); i.e., a multivariate normal distribution with mean zero and variance-covariance proportional to the identity matrix.
Sentence 2: lemma 2.2 The matrix J(f ; x 0 ) is orthonormal for any x 0 , i.e., J T (f ; x 0 )J(f ; x 0 ) = I d . Given the previous condition, model (1) is still not uniquely defined.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In [3] the idea is to introduce a statistics, and large deviation bounds for it, conditioned on sampling from a SBM (with covariates) and on a given C. Experiment Setup Given G, we obtain a clustering C 0 by spectral clustering [15].
Sentence 2: moreover, from Political Blogs data, we see that "smoothing" a graph, by e.g.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For example, (1) has the following CCD approximation However, CCD also introduces strong bias which makes it insufficient to capture the complexity of inferred posterior distributions on real data (see Figure 5).
Sentence 2: consider a parent-child pair in an SBN, S i and S πi , where π i denotes the index set of the parent nodes of S i .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Remark 1 Although we can interpret problem (4) as a MAP estimate of β under appropriate priors when scaling parameters (θ, ρ) are fixed, it does not enjoy an elegant Bayesian interpretation if we perform joint estimation of β and the scaling parameters (ω, ν) because it ignores normalization factors of the appropriate priors.
Sentence 2: although some methods have been developed for either adaptive or multi-task learning, to the best of our knowledge, adaptive multi-task Lasso is the first method that can consider prior information on SNPs and multi-task learning simultaneously in one single framework.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Assume X = Y + ∆ where Y is a nonnegative matrix whose rows have unit 1 norm, where Y = F W is a rank-r separable factorization in which the rows of W are α-robust simplicial, and where ∆ ∞,1 ≤ .
Sentence 2: under this hypothesis, we may also assume that each row of F and of W also sums to one [4].
Relationship: neutral
Domain: NeurIPS
Sentence 1: The entropy condition will be satisfied when log kn k=1 C k card(Ω BI k ) n ε 2 n for some C > 0, where ε n = n −1/2 K f0 log 2β (n/K f0 ).
Sentence 2: when {Ω 0 k } K0 k=1 / ∈ Ω, the complexity number K(f 0 , Ω) can be much larger.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As such, immediate goals for future work include faster algorithm implementations, as well as developing effective postprocessing schemes (e.g., rounding) when the obtained solution is not an exact cluster matrix.
Sentence 2: parameters: Our algorithm has two parameters: c A and c A c . The theorem provides a way to choose their values, assuming we know the values of the bounds p, q.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Therefore applying Talagrand's inequality in the version of [3] to the class G r we obtain Let us define ε r := 30ϕn(r) r + 2τ v nr 2−ϑ 1/2 + 7τ nr .
Sentence 2: now assume that P has Tsybakov noise exponent q ∈ [0, ∞] in the sense of [9].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Lines 14 -18 in Algorithm 1 are added only for the data analysis in Section 5.3, in which p is massive compared to n in that log(p) is comparable to n, and the algorithm may not scale down the size of p sufficiently for even obtaining a ridge regression estimator afterwards.
Sentence 2: the support of β is S = {1, 2, 3, 4, 5}.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (2015) to e O(✏ 3.25 ) for general unconstrained functions, where ✏ is a pre-specifed optimization accuracy; Jain et al.
Sentence 2: for U 2 R m⇥r , ⇧ Orth (U ) returns a matrix U 0 2 R m⇥r that has orthonormal columns.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Here we develop a novel measure, termed the information-theoretic intersection information I II (S; R; C), that quantifies how much of the sensory information carried by a neural response R is used for behavior during perceptual discrimination tasks.
Sentence 2: the redundant information SI(C : {S; R}) is then defined as the solution of the following convex optimization problem on the space ∆ p [20]: where CoI q (S; R; C) ≡ I q (S : R) − I q (S : R|C) is the co-information corresponding to the probability distribution q(s, r, c).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We revisit the notion of policy regret and first show that there are online learning settings in which policy regret and external regret are incompatible: any sequence of play that achieves a favorable regret with respect to one definition must do poorly with respect to the other.
Sentence 2: the distribution ⇡ we want is now the limit of the sequence (⇡) T . The final step is to show that ⇡ is a policy equilibrium.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 2 An advantage of our method is that it allows us to have a different neighborhood size for each data point, depending on the local dimension of its underlying manifold at that point.
Sentence 2: by applying SMCE with λ = 10 on almost 33, 000-dimensional vectorized faces, we obtain a misclassification rate of 2.34%, which corresponds to wrongly clustering 3 out of the 128 data points.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Hence, any predictor w can be written as w = (w i,j ) (i,j)∈G where w i,j is w's weight on φ φ φ i,j (x, y i , y j ).
Sentence 2: the margin Γ(w, x, y) achieved by predictor w at example (x, y) is defined as, We consider the case where the feature map φ φ φ is a potential function for a Markov network defined by a complete graph G with nodes and ( − 1)/2 undirected edges.
Relationship: neutral
Domain: NeurIPS
Sentence 1: It is based on the idea that bandwidth and variable selection can be simultaneously performed by computing the infinitesimal change in a nonparametric estimator as a function of the smoothing parameters, and then thresholding these derivatives to effectively get a sparse estimate.
Sentence 2: the best rate of convergence is n −4/(4+d) , which is impractically slow if d is large.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 1), it refers to restricting the responses or errors that flow in the module from going to infinity, and preventing the network from encountering the vanishing of gradients in the backpropagation process [37].
Sentence 2: overall, both SPNs outperform the other techniques with a significant margin of over 6 intersection-over-union (IoU) points.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Transfer nonadjacencies and endpoint orientations from each G i ∈ G to K and propagate the 3 changes in K using the rules described in [10] PAT({X, Y}, Z) ← all possibly active trails between X and Y given Z for all {X, Y} ⊆ V and The algorithm begins with the complete graph over V with all • endpoints and transfers nonadjacencies and endpoint orientations from each G i ∈ G at line 3, e.g.
Sentence 2: some sequence of edge removals will provably produce H i at line 16 and it will be added to the output set since it is consistent with every G i ∈ G. by theorems 3.1 and 3.2, ION is an asymptotically correct algorithm for learning the complete set of PAGs over V that are consistent with a set of datasets over subsets of V with overlapping variables, if the input PAGs are discovered using an asymtotically correct algorithm that detects the presence of latent common causes, i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Furthermore, the classification boundary occurs uniquely at the point where z = (ŷ, x 1 , ..., x d ), where, from the assumptions, ŷ is the unique solution to (12).
Sentence 2: both z +ε = (y + ε, x 1 , ..., x d ) and z −ε = (y − ε, x 1 , ..., x d ) are, with probability α, on the correct side of the regression surface, defined by z = (ŷ, x 1 , ..., x d ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Bauschke and Borwein [3] give a general condition for the linear convergence of alternating projections in terms of the value  ⇤ (defined in Section 3.1).
Sentence 2: our analysis also shows the effects of additional properties of F . For example, suppose that F is separable, that is, F (V ) = F (S) + F (V \S) for some nonempty S ( V . Then the subsets A rm ✓ V defining the relevant faces of B satisfy either A rm ✓ S or A rm ✓ S c [2].
Relationship: neutral
Domain: NeurIPS
Sentence 1: By reversing the roles of Hand W, the update rule for W can similarly be shown to be nonincreasing .• We have shown that application of the update rules in Eqs.
Sentence 2: a simple additive update for H that reduces the squared distance can be written as (6) If 'flatt are all set equal to some small positive number, this is equivalent to conventional gradient descent.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this section, we discuss concrete families of priors and mechanisms for choosing among them in data-dependent ways.
Sentence 2: one can construct Markov chain algorithms that are geometrically ergodic, in which case n 1/q is replaced by a term 2 W(n) , allowing one to spend computation to control the 1/d 0 term.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Similarly in image restoration, they determine the statistical weights given to the prior knowledge and the received data.
Sentence 2: if we introduce a freezing threshold () so that all spins with (Ui)2 > (}2 are frozen, then the freezing fraction f is given by (11) The thermal average of a dynamic spin in the second stage is related to the cavity fields in both stages, say, for spin 1, where 11,1 is the cavity field in the second stage, r is the order parameter describing the spin correlations of the two thermodynamic stages: i Xtr is the trans-susceptibility which describes the response of a spin in the second stage to variations of the cavity field in the first stage, namely The cavity field 11,1 is a Gaussian variable.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Due to the use of sampled bucket sequences, the time per iteration is nearly independent of the size of the abstraction.
Sentence 2: let σ t i be the strategy used by player i on round t. The average overall regret of player i at time T is: Moreover, define σt i to be the average strategy for player i from time 1 to T . for each information set I ∈ I i , for each a ∈ A(I), define: There is a well-known connection between regret and the Nash equilibrium solution concept.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( 5), we have Now let f satisfy (10) and let x, y ∈ B.
Sentence 2: in particular, they prove that if B = R n , d(x, y) B = x − y 2 we have the gradient characterization They softly enforce this condition by adding a penalty term to the loss function of D that takes the form where the distribution of X is taken to be the uniform distributions on lines connecting points drawn from P G and P r . penalizing the 2 norm of the gradient corresponds specifically to choosing the 2 norm as underlying distance measure on image space.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Any walk between x and y with length , yi with length l, we will show that every node By the definition of G h x,y , v i must be included in G h x,y .
Sentence 2: by extracting a local subgraph around each target link, we aim to learn a function mapping the subgraph patterns to link existence, thus automatically learning a "heuristic" that suits the current network.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Other works in visual control have relied on model free methods that operate on a a low-dimensional state representation of images obtained using autoencoders (Lange et al., 2012;Finn et al., 2016;Kietzmann & Riedmiller, 2009).
Sentence 2: when the number of training examples is increased to 100K both models are at par.
Relationship: neutral
Domain: NeurIPS
Sentence 1: As we have solved for Y * in closed form in Equation ( 8), we can plug this result into Equation ( 1) to obtain a formulation of the problem of optimizing Π * that does not require a simultaneous optimization over Y .
Sentence 2: 0 Given Π We now address the subproblem of solving for the optimal structure matrix, Y , subject only to positive semi-definiteness, for any Π.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this case, the optimization problem associated with computing pred * PD (u) above is a minimum weighted feedback arc set (MWFAS) problem, which is known to be NP-hard.
Sentence 2: recently, Ramaswamy and Agarwal [16] showed the existence of such surrogates, but their result involved an unwieldy surrogate space, and moreover did not give an explicit, usable construction for the mapping needed to transform predictions in the surrogate space back to the original prediction space.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that the case (i) is possible only when the condition in the step (a) holds during the construction of y NEW .
Sentence 2: to derive a contradiction, assume that n t e [1] ≤ n t e [0] and construct a tree-structured GM T e (t) of depth t + 1, also known as the computational tree, using the following scheme: 1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since π is stationary, we have d π P π = d π , and so  The following derives a bound on the Q values as a function of the mixing time.
Sentence 2: their algorithm is not directly applicable to our setting, due to the fact that in our setting, decisions must be made with respect to the current state of the agent (and the reward could be changing frequently), while in their setting the decisions are only made with respect to a single state.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 11 We expect nontrivial interactions between clusters, even when we do not explicitly model them as in [14].
Sentence 2: note that the principle submatrices of a PSD matrix are also PSD; hence, if C ∈ PSD N then C SS , A ∈ PSD K . not only is the recovered A an unbiased estimator for A * M , but also it is now doubly non-negative as A * M ∈ DN N K after the rectification.
Relationship: neutral
Domain: NeurIPS
Sentence 1: samples Refer to the supplementary material for L 2 error bounds and formulas for ν + ε = ν ε (R × Ω) for the gamma, symmetric gamma, and symmetric α-stable processes.
Sentence 2: the learned kernel is then used for GP infilling with 900 training points, taken by down-sampling the training data, which is then applied to the original 44,100 Hz natural sound file for infilling.
Relationship: neutral
Domain: NeurIPS
Sentence 1: PCN-E-5), PCN slightly under-performs ResNet-50 but outperforms ResNet-34, while using fewer layers and parameters than both of them.
Sentence 2: interestingly, the spatial distribution of the prediction error (averaged across all layers) highlights the apparently most salient part of the input image, and/or the most discriminative visual information for object recognition (Fig.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The claim holds for n = 1 as s 1 s −1 The rectangle with side lengths as −1 n and bs n has circumference 2(as −1 n + bs n ) and area as −1 n bs n = ab.
Sentence 2: x 2 +c − 1 for the derivative of φ c . φ c (x) > 0 for c < 0 and φ c is monotonously increasing.
Relationship: neutral