sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: We further assume that the kernel is bounded, that is, |k ψ (x, x')| ≤ M for some M > 0 for all x, x' ∈ X .
Sentence 2: the problem relaxes to the following, Using the Representer theorem, we have that the optimum solution for the above is of the form v * = m i=1 α i ψ(x i ) for some α ∈ R n . Denoting the sample kernel matrix be K such that K i,j = k ψ (x i , x j ), the above optimization problem is equivalent to the following optimization problem, where Y is the vector corresponding to all y i and ||Y || ∞ ≤ 1 since ∀i ∈ [m], y i ∈ [0, 1].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Formally, on each iteration T + 1, player i selects actions a ∈ A(I) according to probabilities If a player plays according to CFR in every iteration, then on iteration T , R T (I) So, as T → ∞, In two-player zero-sum games, if both players' average regret [15].
Sentence 2: cFR constitutes an anytime algorithm for finding an -Nash equilibrium in zero-sum games.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The fact that F µ ( ) is indeed smooth with modulus µ follows from Lemma 6.1 in the Supplement.
Sentence 2: one is able to apply an accelerated gradient descent algorithm on this modified Lagrange dual function, which is detailed in Algorithm 1 below, starting from an initial primal-dual pair (e x, e ) 2 R d ⇥ R N . Algorithm 1 Primal-Dual Smoothing: PDS ⇣ e , e x, µ, T Our proposed algorithm runs Algorithm 1 in multiple stages, which is detailed in Algorithm 2 below.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Clearly, In a similar way we can write, L p K φ L,2 0 (x) = π 2 λ 2 0 φ L,2 0 + T 2 (x) where, T 2 (x) = π1 π2λ 2 0 k(x, z)φ L,2 0 (z)p 1 (z)dz.
Sentence 2: when T 1 (x) and T 2 (x) are small enough then φ L,1 0 and φ L,2 0 are eigenfunctions of L p K with corresponding eigenvalues π 1 λ 1 0 and π 2 λ 2 0 respectively.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The sample complexity of an algorithm A is the function m A defined as follows: For , δ > 0, m A ( , δ) is the smallest integer such that for every m ≥ m A ( , δ) and every distribution D on X × Y, with probability of > 1 − δ over the choice of an i.i.d.
Sentence 2: sample S of size m, The first term on the right-hand side is the approximation error of H. the sample complexity is the number of examples required to ensure that the estimation error of A is at most (with high probability).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Following (21), the gradient of the KL divergence functional F (q) := KL(q || p) is grad H KL(q || p) = ∇ • (φ * q,p q).
Sentence 2: the SVGD-Valsov equation ( 13) is a gradient flow of KL divergence under metric W H (•, •): ∂q t ∂t = −grad H KL(q t || p).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a MDP M and a homomorphism, an aggregated MDP M ′ is given by (S ′ , A, {P(C, a, D) : a ∈ A;C, D ∈ S ′ }, {R(C, a) : a ∈ A,C ∈ S ′ }, ρ, g s : s ∈ S) where S ′ is a partition of S, ρ : S → S ′ maps states to their aggregates, each g s : A → A relabels the action set and we have that ∀C, D ∈ S ′ and a ∈ A, Note that all the states in a partition have actions that are relabelled specifically so they can exactly match each other's behaviour.
Sentence 2: a policy in the aggregate MDP can be lifted to the original MDP by using this relabeling.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We provided a key result stating that a set can be approached to a given precision using only a finite number of steering policies, which may be learned on-line.
Sentence 2: an interesting observation regarding the proposed learning algorithm is that the learned optimal polices in each direction are essentially independent of the target set T . the target set need not be fixed in advance and may be modified on-line without requiring a new learning process.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: SURE-T methods yielded a slightly better denoising performance than SURE-based methods (approximately 0.02 -0.06 dB and 0.04 -0.12 dB for BSD68 and Set12 datasets, respectively) with considerably increased the overall inference time.
Sentence 2: at this moment, SURE-T methods do not seem to have considerable benefits compared to SURE-based methods.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This constraint on the entropy is crucial since otherwise the solution of the LMO would be a degenerate distribution as the authors of [5] have also argued.
Sentence 2: we now replace this problem with its regularized form using Lagrange multipliers and solve for s given a fixed value of : the regularized LMO problem is equivalent to the following minimization problem: where Z is the normalization constant of q p q t . From this optimization problem, we can write what we call the Residual Evidence Lower Bound (RELBO) as: Discussion.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, there is no way to invert the modeling process to determine parameter ranges given a fixed sum of squared error.
Sentence 2: we use the algorithm detailed in §2 to map out the confidence surface as a function of the input parameters; that is, we use the algorithm to pick a location in the seven dimensional parameter space to perform an experiment, and then run CMBFast [12] to create simulated power spectrum given this set of input parameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, some regions outside the grid cells are also raised, but were supposed to be zero.
Sentence 2: we address this issue via another, separate level set property: there is a threshold T such that a) the function value inside each I δ k is larger than T ; b) the function values outside the grid cells are smaller than T . the desired grid indicator function can be obtained by performing a max operator with the threshold T , i.e., cutting off the smaller values and setting them to zero (see Appendix C).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since ρ is continuous, V is the image of a compact set under a continuous map, and thus V is compact and the inverse of the bijective map I • ρ : X → W is continuous.
Sentence 2: there is a one-to-one relationship between the continuous functions f X on X and the continuous functions f W on W , namely C(X) = C(W ) • I • ρ, see also the discussion following (7).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This observation parallels one of the key insights of [10], here applied to M instead of K. Unfortunately, even though we can pose a convex objective, it does not allow us to immediately solve our problem because we still have to relate M to y, and M = yy is not a convex constraint.
Sentence 2: the main challenge is to find a way to constrain M to ensure M = yy while respecting the class balance constraints − ≤ e y ≤ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This bound will depend on the empirical risk R S (G Q ) which relates to the the empirical loss ζ Q (measured on the training sequence S of m examples) through the equation where Note that Equations 7 and 8 imply that Hence, any looseness in the bound for R(G Q ) will be amplified by the scaling factor c on the bound for ζ Q .
Sentence 2: within this approach, the bound for ζ Q can be tight only for small values of c. Note however that loss functions having a small value of c are commonly used in practice.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This allows to reparameterize w qm according to w qm = s qm w qm and assign the above prior distributions on s qm and w qm .
Sentence 2: the reparameterized spike and slab prior takes the form: Notice that the presence of w qm in the likelihood function in (1a) is now replaced by the product s qm w qm .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: our goal -to synchronize topics across different languages -and there's no reason to believe that likelihood measures would assess that: a model where topics are synchronized across languages is not necessarily more likely than a model that is not synchronized.
Sentence 2: to evaluate our model we compare the topic distributions of each English document with its corresponding French pair (and analogously for the other combinations: English/Portuguese and French/Portuguese), with these metrics: where L 1 denotes the set of documents in the first language, F a mapping from a document in the first language to its corresponding translation in the second language and θ d the topic distribution of document d. Mean Hellinger distance: Agreements on first topic: where I is the indicator function -that is, the proportion of document pairs where the most likely topic is the same for both languages.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 1 1/ poly(n), the algorithm can compute marginal contribution estimates ṽ that are all a 1/ poly(n)-additive approximation to the true marginal contributions v (See Appendix B for formal analysis of estimates).
Sentence 2: we give the analysis for the true marginal contributions, which, with probability 1 1/ poly(n) over the samples, easily extends for arbitrarily good estimates.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In fact, any minimum of (1) will be such that rank[Z] + max j e •j 0 ≤ n and rank[Z] + max i e i• 0 ≤ m, regardless of Y. Secondly, unlike any separable penalty function (4) that retains the correct global optimal as (1), Proposition 2 implies that (14) need not be locally minimized by every possible support pattern for outlier locations.
Sentence 2: premature convergence to suboptimal supports need not disrupt trajectories towards the global solution to the extent that (4) may be obstructed.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As this formulation only considers real-valued cross spectral densities, the authors propose a multivariate covariance function by including a complex component to the cross spectral densities to cater for phase differences across channels, which they call the Cross Spectral Mixture kernel (denoted CSM).
Sentence 2: this multivariate covariance function can be seen as the proposed MOSM model with µ i = µ j , Σ i = Σ j , θ i = θ j ∀i, j ∈ {1, . . . , m} and φ i = µ i ψ i for ψ i ∈ R n . the SM-LMC is a particular case of the proposed MOSM model, where the parameters µ i , Σ i , θ i are restricted to be same for all channels and therefore no phase shifts and no delays are allowed-unlike the MOSM example in Fig.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In fact, if f ✓ is a bijection between samples in high probability regions, then f 1 ✓ is automatically also such a bijection.
Sentence 2: we ignore f 1 ✓ during training and only train f ✓ (x, v) to reach the target distribution p(x, v) = p d (x)p(v).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, if we consider our algorithm to take samples in blocks of size Poi(m), the probability that we have not found an r-wise collision after t blocks is at most (1 − c r ) t . Thus, the expected number of blocks until we have an r-wise collision is O r (1).
Sentence 2: the expected number of samples is O r (m) = O r (1/ p r ) completing the proof.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: (Sketch) For each ℓ = 1, . . . , n, observe that T ℓ is x ℓ -tight.
Sentence 2: we can show by induction that any cluster in P ℓ is x ℓ -tight for each ℓ = 1, . . . , n. f λ The procedure FINDPARTITION(λ) returns h λ ∈ R and P λ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The components of , defined in this way, are Gaussian basis vectors slightly narrower than those in £ (this is easily seen by noting that the eigenvalue spectrum for the Toeplitz matrix £ is given by the Fourier transform, and that the square-root of the Gaussian function in the Fourier space is a Gaussian of larger width, corresponding to a smaller width in the original space).
Sentence 2: weight vectors obtained through ARD The results of this procedure (labelled ASD/RD) on our example recording are shown in the rightmost panel of figure 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: These functions can be shown to be δ-related as well as λ-strongly convex.
Sentence 2: moreover, the optimum of F (w) = 1 2 (F 1 (w) + F 2 (w)) equals we see that the optimal point w * depends on the j-th column of M . Intuitively, the machines need to approximate this column, and this is the source of hardness in this setting: Machine 1 knows M but not j, yet needs to communicate to machine 2 enough information to construct its j-th column.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Otherwise, the buyer is largely indifferent to being offered prices p or p , but distinguishing between the two prices is essential for the seller during exploit rounds.
Sentence 2: we assume that the value distribution D is K-Lipschitz, which eliminates this problem: Defining F (p) ≡ Pr v∼D [v ≥ p], we assume there exists K > 0 such that This assumption is quite mild, as our Phased algorithm does not need to know K, and the dependence of the regret rate on K will be logarithmic.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, during the ascent from vertex v k ∈ V m to the root we will compute {r The computational complexity is obtained by noting that every ascent to the root requires O(S T ) steps and along each ascent we must compute up to max(m, S T ) resistances.
Sentence 2: the total complexity is O(m 2 + mS T ), assuming that each step of the algorithm is efficiently implemented.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If the r(•) is non-negative, it is easy to see that this diversity function is monotone and submodular -adding a new window never hurts, and since the marginal gain is the score contribution of the new regions not covered by previous window, it is naturally diminishing.
Sentence 2: even though this heuristic not was presented as such, the authors of [26] did in fact formulate a near-optimal greedy algorithm for maximizing a monotone submodular function.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let's look at the gradients of L q loss to see this: where f yi (x i ; ✓) 2 [0, 1] 8 i and q 2 (0, 1).
Sentence 2: relative to CCE, L q loss weighs each sample by an additional f yi (x i ; ✓) q so that less emphasis is put on samples with weak agreement between softmax outputs and the labels, which should improve robustness against noise.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By definition of I , we know r ∈ I =⇒ V (r, y 1 ) < 0 < V (r, y 0 ) and the orientation of V then implies Γ(δ y1 ) > r > Γ(δ y0 ).
Sentence 2: γ(δ y1 ) = γ(1) ≥ b > a ≥ Γ(δ y0 ) = γ(0), with the strict inequality since I is nonempty.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When T = ( Ṽ , Ẽ) is a part of a "larger" tree T = (V, E), i.e., Ṽ ⊆ V and Ẽ ⊆ E, we can treat T as a "super-node" of the tree T with value m( T ).
Sentence 2: we have the following definition of a super-tree (note that a super-tree provides a disjoint partition of the given tree): Definition 5.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let ǫ > 0 be given, and let (f, g) be a selective regressor chosen by Strategy 1 after observing the training sample S m . Let f be an ERM.
Sentence 2: then, We thus have, by the definition of fǫ,x , and using the convexity of ℓ, together with Jensen's inequality, As for the other direction, if So far we have discussed the case where ǫ is given, and our objective is to find an ǫ-pointwise optimal regressor.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We would like to optimize the margin by solving the optimization problem max Introducing one additional variable ζ k per classifier, problem (1) is equivalent to: Considering the unnormalized rows W k /ζ k , we obtain the following equivalent formulation: When y = −1 and z(x i ) satisfy the margin constraint in (3), we have that the constraint holds for every sub-classifier k since y i W k x i is minimal at k = z(x i ).
Sentence 2: when y = −1, we can enforce the constraint for all k. We can also smooth the objective into a convex, defined everywhere one by minimizing the sum of the inverse squares of the terms instead of maximizing the sum of the terms.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, it can be checked that for a, b ∈ A, we have L(V, a)/ L(V, b) = i∈{2,...,m} (p i /p 1 ) SC APP (b)−SC APP (a) .
Sentence 2: sC APP (a) ≥ SC APP (b) if and only if L(V, a) ≥ L(V, b), as required.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Their algorithm can also be formulated into the statistical query model; we use the simpler version in (3.16) for ease of presentation.
Sentence 2: with more sophicated proof techinique, it can be shown that � s 2 /n ∧ (1/α 2 • s log d/n) is the critical threshold for asymptotically powerful detection with computational efficiency.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , b n are different, the polynomial c ⊤ ν n (x) has no repeated factor.
Sentence 2: there is a δ > 0 and a T > 0 such that for all t > T we have Dν n (x(t)) ⊤ c ≥ δ and Dν n (x(t)) ⊤ ĉ(t) ≥ δ (see proof of Theorem 3 in [9] for the latter claim).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By adding the regularization , and substituting G = X T B, we get the regularized where XDX T , and η is the positive weighting factor.
Sentence 2: different from the existing regularizations [8,9,14] for SIR, which are constructed as deterministic terms, the manifold regularization in ( 6) is a random term that involves two data dependent variables (matrices) Q and S. it is necessary to discuss the convergence property of the proposed manifold regularization.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For η = 1 (homogeneous models) and regular graphs ∆ min = ∆ max = ∆, the sample complexity simplifies to n = Ω ∆ 2 ρ −2 (log p) 3 .
Sentence 2: we can characterize an explicit dependence on the fraction of observed nodes ρ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To arrive at a sparse approximation of the stimuli either one of the dictionary elements could potentially be used, but due to the high cost of non-sparseness both of them together are not likely to be chosen a given representation.
Sentence 2: small changes in the image, as might arise due to various noise sources, might cause one of the coefficients to be preferred over the other in an essentially random fashion, potentially resulting in very different coefficient values for highly similar images.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We next consider the posterior on the track meta-states: i,k (s where (18) follows from (2).
Sentence 2: if P (s i,• ) follows a Markov chain then the form for q(s i,• ) is the same as a hidden Markov model (HMM) with emission log likelihoods i,k ∈ [R − ] N S . the meta-state posterior q(s i,• ) update is implemented using the forward-backward algorithm [21].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To show this, we express the Fisher discriminant ratio f as the composition where g(u, t) = u 2 /t and H is the function The function H is linear (as a mapping from µ x , µ y , Σ x , and Σ y into R 2 ), and the function g is convex (provided t > 0, which holds here).
Sentence 2: the composition f is a convex function of µ x , µ y , Σ x , and Σ y .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We employ RJ-MCMC, which extends the capability of conventional MCMC to allow sequential samples of different dimensions to be drawn (Green, 2003).
Sentence 2: a posterior distribution is not limited to coefficients and other parameters of a fixed basis expansion, but can represent a changing number of basis functions, as required by the description of Lévy processes described in the previous section.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The only difference is the factor of n in the denominator in the case of ARD, but this can be offset by an appropriate rescaling of the FOCUSS λ trade-off parameter (analogous to Σ ).
Sentence 2: aRD can be viewed in some sense as taking the recursive FOCUSS update rules and including the sLORETA normalization that, among other things, allows for depth bias compensation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Geodesics on D are realized as exponential maps from tangent spaces to D. The intricate geometry of D disallows explicit analytic expressions.
Sentence 2: we adopt an iterative strategy, where in each step, we first flow infinitesimally in the prescribed tangent direction in the space L 2 , and then project the end point of the path to D. Next, we parallel transport the velocity vector to the new point by projecting the previous velocity orthogonally onto the tangent space of D at the new point.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this step of the proof we generalize the technique of [19] to multi-kernel classes.
Sentence 2: first we note that, since the (centered) covariance operator E φ m (x) ⊗ φ m (x) is also a self-adjoint Hilbert-Schmidt operator on H m , there exists an eigendecomposition is an orthogonal basis of H m . Furthermore, the no-correlation assumption (U) entails E φ l (x) ⊗ φ m (x) = 0 for all l = m. for all j and m, Let now h 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The marginal of λ over the initial state is the initial state distribution, so the second term of left hand side, s v(s)η(s), converges to E π i ,π −i [r i ], which is the second term of right hand side.
Sentence 2: the left hand side and right hand side of Equation ( 8) are the same as t → ∞.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Even though this pairwise correlation clustering takes a rich pairwise feature vector and a trained parameter vector (which will be presented later), it often produces incorrectly predicted segments due to the segment boundary ambiguities caused by limited pairwise relations of neighboring superpixels (see Figure 2).
Sentence 2: to incorporate higher-order relations, we develop higher-order correlation clustering by generalizing the correlation clustering over a hypergraph.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since P contains at most B solutions for each possible size {0, . . . , 2k − 1}, |P | ≤ 2Bk.
Sentence 2: j max can increase by at least 1 in one iteration with probability at least , where 1 |P | is the probability of selecting x in line 3 of Algorithm 3 due to uniform selection and 1 n (1 − 1 n ) n−1 is the probability of flipping only a specific bit of x in line 4.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this semi-autonomous local government, county managers act as executives, and the individual departments are synonymous with the individual departments and agencies in, for instance, the U.S. federal government.
Sentence 2: not only does this email data set offer a view into the communication patterns of the managers of New Hanover County, but analyses of it also serve as case studies in modeling inter-agency communications in the U.S. federal government administration.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For the 148/176 images with faces, SM+VJ was better than SM alone for 144/148 images (p < 10 −29 ), whereas VJ alone (equal to the face conspicuity map) was better than SM alone for 83/148 images, a fraction that fails to reach significance.
Sentence 2: although the face conspicuity map was surprisingly predictive on its own, fixation predictions were much better when it was combined with the full saliency model.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If sample size n > c(max i w 2 (C i ∩ S p−1 ) + log k)/ρ 2 , then we have with probability at least for constants c, C > 0 that depend on sub-Gaussian norms |||Z||| φ2 and |||ω||| φ2 .
Sentence 2: assuming the SC condition in ( 9) is satisfied, the sample complexity and error bound of the estimator depends on the largest Gaussian width, rather than the sum of Gaussian widths.
Relationship: reasoning