sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Note that this exponentially large set of linear constraints can be replaced by a single, equivalent, non-linear constraint: In a factored MDP, the reward function is represented as the sum of local rewards 0 R A R . Furthermore, the basis B C and its backprojection R V are also functions that depend only on a small set of variables.
Sentence 2: the right side of the constraint can be viewed as the sum of restricted scope functions parameterized by p . For a fixed p , we can compute the maximum over I % using a cost network, as in Section 2.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: negative) labeled points tend to co-vary.
Sentence 2: there is a reason to prefer the GP with a graph kernel, and we do expect humans to learn on manifold in 4 l moons U h. We also explore the convex combination of the seven GPs as a richer model for human behavior: k(λ) = 7 i=1 λ i k i , where λ i ≥ 0, i λ i = 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Equipped with the results in Theorem 2 and Lemma 3, we can now show that the fixed point solution of T[V ](x, y) = V (x, y) is unique, and equals to the solution of the CVaR MDP problem (3) with x 0 = x and α = y. Theorem 4 (Optimality Condition) For any x ∈ X and y ∈ (0, 1], the solution to T[V ](x, y) = V (x, y) is unique, and equals to Next, we show that the optimal value of the CVaR MDP problem (3) can be attained by a stationary Markov policy, defined as a greedy policy with respect to the value function V * (x, y).
Sentence 2: while the original problem is defined over the intractable space of history-dependent policies, a stationary Markov policy (over the augmented state space) is optimal, and can be readily derived from V * (x, y).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In (ii) & (iii), negating any ρ ij , it still holds that J = I − R 0 : (ii) negating ρ ij doesn't affect the eigenvalues of J (remove edge {i, j} and, in each eigenvector, negate all entries in one subtree); (iii) negating ρ ij preserves J {i,j} 0 in (4) so J 0.
Sentence 2: making all ρ ij > 0, we find I − R 0 and R ≺ I.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this case, the loss term in ( 6) is simply kY W H T k 2 F . Thus, setting rf (H) = 0 is equivalent to solving the following Sylvester equation for an n ⇥ k matrix H: (7) admits a closed form solution.
Sentence 2: however the standard Bartels-Stewart algorithm for the Sylvester equation requires transforming both W T W and L h into Schur form (diagonal in our case where W T W and L h are symmetric) by the QR algorithm, which is time consuming for a large L h . we consider applying conjugate gradient (CG) to minimize f (H) directly.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The location of the sound source was limited to be in front of the head (¥ © © ¢¡ "¦£ # U# V# ¡ "¦£ ) at a fixed constant elevation and in jumps of I §¤£ .
Sentence 2: the number of classes, © , for training is I &¡ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This term appears in F, either directly or in expectations, in terms t through T of the summation: steps on which q(zt|x ≤t , z<t) depends terms in which q(zt|x ≤t , z<t) appears However, the filtering setting dictates that the optimization of the approximate posterior at each step can only condition on past and present variables, i.e.
Sentence 2: steps 1 through t. of the T terms in F, the only term through which we can optimize q(z t |x ≤t , z <t ) is the t th term: q * (z t |x ≤t , z <t ) = arg min q(zt|x ≤t ,z<t) Ft . Optimizing Ft requires evaluating expectations over previous approximate posteriors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: between expanding a leaf following a path that is optimistic according to ρ+ t (step 7,8,9), or according to ρ− t (step 11.)
Sentence 2: at a high level, the algorithm performs on each cell (h, i) ∈ P of the given partition two hierarchical searches, one for the maximum value of ρ in region R(h, i) and one for its minimal value.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is easy to understand why this occurs by considering the prior distribution over amplitude and phase implied by our choice of prior distribution over x t (or equivalently over z t ), Phase and amplitude are dependent in the implied distribution, which is conditionally a uniform distribution over phase when the amplitude is zero and a strongly peaked von Mises distribution [15] when the amplitude is large.
Sentence 2: the model favors more highly variable IFs at low amplitudes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If our proposal is deterministic, then f ✓ (f ✓ (x, v)) = (x, v) should hold for all (x, v), a condition which is difficult to achieve 2 .
Sentence 2: we introduce a proposal which satisfies Equation ( 9) for any ✓, while preventing random walk in practice by resampling v after every MH step.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, using the techniques in [10] it is straightforward to show that (11) is equivalent to a * (λ) ≤ c 2 λ 2β 1−β .
Sentence 2: if H approximates P with exponent β then the rate in Theorem 3.1 becomes n − 4β (2+p)(1+β) which is the rate we established in Theorem 2.3 for the original SVM.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Throughout this section, we assume that F is a non-negative submodular set-function that is equal to zero at ∅ and V . This immediately implies that f is invariant by addition of any constant vector (that is, f (w + α1 V ) = f (w) for all w ∈ R p and α ∈ R), and that f (1 V ) = F (V ) = 0.
Sentence 2: contrary to the non-decreasing case [3], our regularizers are not norms.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the kernel calculation with the statistical feature selection, if χ 2 (u) < τ holds, that is, u is not statistically significant, then u is eliminated from the features, and the value of u is presumed to be 0 for the kernel value.
Sentence 2: the sequence and tree kernel with feature selection (SK,TK+FS) can be defined as follows: (2) The difference with their original kernels is simply the condition of the first summation, which is τ ≤ χ 2 (u).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Accumulated till d max basis functions are added, this leads to a T selection that has O(n 2 d 2 max ) complexity, which is disproportionately higher than T basic .
Sentence 2: smola and Bartlett (2001) resorted to a randomized scheme by considering only κ basis elements randomly chosen from outside I during one basis selection.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From Example 3 and the proofs of Theorems 2 and 3 we can see that the non-submodularity only occurs in the case when two or more cascades arrive at one node at the same time.
Sentence 2: if such a scenario does not happen frequently, the Max-M and Min-M problems will be close to submodular optimization problems, and consequently, the greedy algorithm is effective.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using the additive property of the ΓP, posterior sample for the sum of λ k over unused atoms is obtained as .
Sentence 2: we obtain a closed-form posterior sampler for the rate parameter c 0 of the ΓP as c 0 | − ∼ Gamma(e 0 + γ 0 , f 0 + λ γ0 + K+ k=1 λ k ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 1 to zero we find that E q * [(V * 0 − V ) K ] = 0, where q * ≡ q( • ; λ * ) is the variational distribution at the optimum.
Sentence 2: the lower bound at the optimum is , where the sum runs only to K − 1 because the term with k = K vanishes at V 0 = V * 0 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this framework, we generate deep GNNs by "stacking" GNN layers in a hierarchical fashion (Figure 1): the input nodes at the layer l GNN module correspond to the clusters learned at the layer l − 1 GNN module.
Sentence 2: each layer of DIFFPOOL coarsens the input graph more and more, and DIFFPOOL is able to generate a hierarchical representation of any input graph after training.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The 2N distances can determine the location of the vertices along the dimension up to a shift, through use of the law of cosines: For each subsequent dimension, two new vertices are chosen and new dissimilarities (δ ai , δ bi ) are computed by Dijkstra's algorithm.
Sentence 2: the subsequent dimensions are assumed to be orthogonal to previous ones, so the distances for dimension N are computed from the dissimilarities via: each dimension accounts for a fraction of the dissimilarity matrix, analogous to PCA.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This is because b > 0 for all integers k ≥ 1 with 4k ≤ M . So choosing M = 4k, k ≥ 1, leads to a pathological approximation of φ logit where the log-likelihood can be made arbitrarily large by taking θ 2 → ∞.
Sentence 2: a reasonable polynomial approximation for logistic regression requires M = 2 + 4k, k ≥ 0.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: What makes it slightly unusual and perhaps of more general interest is the fact that our sensor data are recorded as averages over each minute (our analysis is off-line, for the purpose of making recorded data useable for biomedical research), whereas the events of interest-in this case, re-zeroings, line flushes, and blood draws-can occur at any time and have durations ranging from under 5 seconds to over 100 seconds.
Sentence 2: the natural time step for modeling the sensor state transitions might be one second, whereas the measurement interval is much larger.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Remark Because g(x, y) = ∇ x log p(x)k(x, y) + ∇ x k(x, y), and ∇ x log p(x) is often unbounded if the domain X is not unbounded.
Sentence 2: the condition that g(x, y) must be bounded in Lemma 3.1 suggests that it can only be used when X is compact.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The kernel for Ĝr = I and F = r F r (whose entries are f ij ) should be positive semidefinite [15].
Sentence 2: choosing F = 11 T (all entries 1) and multiplying the kernel by 1 and using K RBF as the kernel on vectors, we get back the kernel defined in (1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: After establishing that δ is an unbiased estimator, we show that the probability of a large deviation between δ and δ decreases exponentially in |V − L|, the number of internal nodes of T δ .
Sentence 2: to show a high probability bound for our size estimate, we need to establish a lower bound on the number of internal nodes of T δ , which will be the final step in the analysis.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: A multi-class binary decision tree in dimension d can be defined by a pair (t, h), where t is a binary tree with a variablethreshold question at each internal node, e.g., X j ≤ θ, j ∈ [1, d], and h = (h l ) l∈Leaves(t) a vector of distributions over the leaves Leaves(t) of t. At any leaf l ∈ Leaves(t), h l (y) ∈ [0, 1] for all y ∈ Y and y∈Y h l (y) = 1.
Sentence 2: for convenience, we will denote by t(x) the leaf l ∈ Leaves(t) associated to x by t. the score associated by (t, h) to a pair (x, y) ∈ X × Y is h l (y) where l = t(x).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For simplicity, let us assume that kl/δ = O(1) and ε = o(1).
Sentence 2: by Lemma 5.2 we know that unless v ≤ ε, there are Ω(1) fraction of r such that ∂L(w)/∂w r 2 = Ω(ε).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this case, we exactly solve the asymptotic dynamics and prove that w(t) = log(t) ŵ + O(1) where ŵ is to the L 2 maximum margin separator.
Sentence 2: the margin (affecting generalization) improves slowly (as O(1/ log(t)), even while the training error is very low.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We show that, for all odd K, L (K) (λ, V 0 ) is a nontrivial lower bound on the marginal likelihood p(x).
Sentence 2: we propose the perturbative black box variational inference algorithm (PBBVI), which maximizes L (K) (λ, V 0 ) over λ and V 0 with stochastic gradient descent (SGD).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In other words, if the summary S contains private data of a user v = u, the user u cannot have access or evaluate v's private part of S, i.e., S ∩ V v . In public-private data summarization, we would like to find the smallest subset S ⊆ V such that all users reach a desired utility A naive way to solve the above problem is to find a separate summary for each user and then return the union of all summaries as S. A more clever way is to realize that problem (1) is in fact equivalent to the following problem [13,14] OPT = arg min Note that the surrogate function f (•) is also monotone submodular as a thresholded submodular function remains submodular.
Sentence 2: finding a set S that provides each user with utility Q is equivalent of finding a set S with f (S) ≥ L . = Q × |U|.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As a result, we can approximate (M u − M u D(λ)M u ) −1 as follows: .
Sentence 2: we can write ω(f l , ρ) as follows: where φ(f l , ρ) is the output of the following optimization problem max We can simplify the above problem by approximating As the above problem is a linear programming problem, the solution for λ can be computed as: From the above formulation, we find that ρ plays the role of a threshold of selecting the unlabeled examples.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Proposition 15 Consider the set of points that are solutions of Eq.
Sentence 2: 3 for some seed For the reverse direction, suppose that w * ∈ W G , in which case there is some v ∈ V such that w * G v and L(w * ) ≤ L(w) for any w G v. Since G is a pre-order, it must hold in particular that L(w * ) ≤ L(w) for any w G w * G v. we also have that w * ∈ arg min w G w * L(w), i.e., w * ∈ W G . We describe the results of numerical experiments when regularizing with the permutahedron (symmetric group) and the signed permutahedron (hyperoctahedral group).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that the bounds generally depend on distribution-specific properties of the Markov chain (e.g., P , t mix , γ ⋆ ), which are often unknown a priori in practice.
Sentence 2: much effort has been put towards estimating these unknown quantities, especially in the context of MCMC diagnostics, in order to provide datadependent assessments of estimation accuracy [e.g., 11,12,[17][18][19].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For each cluster k, we independently drew M (k) * samples from N H (k) * (0, 10I H (k) * ), where N d (µ, Σ) denotes the d-dimensional Gaussian, and projected them into the observed L-dimensional space by , each entry of which follows N 1 (0, 1).
Sentence 2: we obtained a noiseless matrix for the k-th cluster.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, by standard concentration argument our test statistic Note that the latter term is determined by the angle of θ(M :t , U k ).
Sentence 2: our test statistic in Step 2 is indeed an effective measure of θ(M :t , U k ), or θ(L :t , U k ) since L :t ≈ M :t , as proven by the following novel result.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We now sketch the proof of Theorem 1.1; almost all the details are relegated to the appendix.
Sentence 2: recall that for n ≥ n o , we take F n to be the sigma-field of all outcomes up to and including time n, that is, An additional piece of notation: we will use u to denote u/ u , the unit vector in the direction of u ∈ R d . for instance, the Rayleigh quotient can be written G(v) = v T A v. We first bound the expected improvement in Ψ n in each step of the Krasulina or Oja algorithms.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that switching to a motion path without any motion constraint (equation ( 2) with WI = 0) will never cause any additional discontinuities, because the smoothness constraint is the only one left.
Sentence 2: we solve our problem by precomputing a cost function that does not constrain the motion for a couple of transitions, and then starts to constrain the motion with the new motion direction.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, assume grandpaOf, ABE and BART represent the indices of the respective symbols in a global symbol vocabulary.
Sentence 2: then, the new proof state constructed by unify is: the output score of the neural network S 0 ⇢ will be high if the subsymbolic representation of the input s is close to grandpaOf and the input i is close to BART.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As shown in [9], PCA reveals that the Origin-Destination (OD) flow matrices of ISP backbones have low intrinsic dimensionality: For the Abilene network with 41 links, most data variance can be captured by the first k = 4 principal components.
Sentence 2: the underlying normal OD flows effectively reside in a (low) k-dimensional subspace of R n . This subspace is referred to as the normal traffic subspace S no .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To compute Q w (z, s) for a rational loss, recall that Q w (z, s) ∝ y : y s−q+1:s =z e L U (y,yi)+w•Ψ (xi,y) .
Sentence 2: we will design two WFAs, A and B, such that A(y) = e w•Ψ(xi,y) , B(y) = e L U (y,yi) , and their composition C(y) = (A • B)(y) = e L U (y,yi)+w•Ψ(xi,y) .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let f w (x) be the function computed by a d layer feed-forward network with parameters w and Rectified Linear Unit (ReLU) activations, where φ(z) = max{0, z}.
Sentence 2: let h i be the number of nodes in layer i, with h 0 = n. for any layer i, we have W i ∈ R hi×hi−1 .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [12] showed that such networks can express any function computed by a Turing machine, similar to networks with ReLU or sigmoid activations.
Sentence 2: following [12], we consider the class of 2-layer polynomial networks with inputs from R d and k hidden neurons: If we write A = k i=1 a j w j w j , we have the following equivalent formulation: if replace the hard rank constraint with trace norm A * ≤ θ, the task of empirical risk minimization (ERM) given training data {(x 1 , y 1 ), .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We introduce the learning the learning rate (LLR) algorithm, which satisfies (2), ( 3) and ( 4) and in addition guarantees a regret satisfying for any ε > 0.
Sentence 2: lLR performs almost as well as the learning rate ηT ∈ [η ah t * , 1] that is optimal with hindsight.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Similarly, we assume that the distribution of features of different phones (i) is a Gaussian distribution with covariances ~i' Therefore, h(X/Y1) = ~ l: p (y~)Iog (2net I~il (6) yiCYi Finally, we assume that the distribution of features of different phones spoken by different speakers is also a Gaussian distribution with covariances ~ij.
Sentence 2: h(X/Y1,Y2 ) = ~ l: p(yLYOlog(2netl~ijl (7) y;CY1,y~CY2 Substituting equations 6 and 7 in equations 4 and 5, we get J(X ' Y;)=~lo I~I .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that f l i is a sub-vector of f l , so equation ( 8) can be written in a compact form as: where o l and f l are the same as in (3), while the matrix A = [a ij ] ∈ R n×n is constructed as follows: ∀x i and x j , 1 ≤ i, j ≤ n, if x j ∈ N i , then a ij equals the corresponding element of α i in ( 9), otherwise a ij equals 0.
Sentence 2: similar as α i , the matrix A is also independent of f l and the cluster index l. Substituting (10) into (3) results in a more specific optimization problem: subject to F is a scaled partition matrix (12 where based on the KRR algorithm, we have transformed the objective function (3) into the quadratic function (11).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: -Type II: Consider a strict KL contraction operator with β = 1, denoted as Φ t , is parameterized by an auxiliary parameter t that is different from θ, and for any distribution p ∈ Ω d , we have Φ 0 {p} = p and Φ t {p} is continuously differentiable with regards to t. Then, the KL divergence Φ t {p} and Φ t {q θ } can be regarded as a function of t and θ, as: L(t, θ) = KL(Φ t {p} Φ t {q θ ).
Sentence 2: the KL contraction in Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: [4] has shown that the PŁ condition is the weakest condition among the following conditions: strong convexity (SC), essential strong convexity (ESC), weak strong convexity (WSC), restricted secant inequality (RSI) and error bound (EB).
Sentence 2: the convergence analysis for the PŁ condition naturally extends to all the above conditions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Such plasticity may disrupt the recall of stored episodic patterns by changing the correspondence between the input areas and EP.
Sentence 2: if the brain is still to be able to recall hippocampally stored episodes, it either needs to maintain the correspondence between the low-level and EP representations of the episodes by restricting neocortical learning (achieved in the previous model by having the hippocampus replay its old episodic patterns along with the new semantic patterns governing continued neocortical plasticity), or it needs to update the connections between the hippocampus and EP such that the hippocampally stored pattern continues to match the EP representation of the input pattern corresponding to the episode.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, for this case, it is easily checked that any subgraph of G obtained by deleting exactly one edge results in a (k − 1) tree, and every sub (k−1)-tree results from this operation.
Sentence 2: the optimal (k−1)-tree can be found using Algorithm 1, and in this case, the complexity of Algorithm 1 is O(n(k + 1) 2 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In our setting, where the adversary chooses a different entry at each round, [21,Theorem 6] implies that for the class W of all matrices with trace-norm at most r = O(n), it holds that R T (W )/T ≤ O(n 3/2 /T ).
Sentence 2: r n 2 (W ) ≤ O(n 3/2 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, though theoretical bound on the pre-constant ζ(ε, ρ) as function of 1/ε and ρ is not very exciting, our simulations suggest (see Section 5) that even for hard problem setup, the performance is much more optimistic than predicted by these theoretical bounds.
Sentence 2: as a recommendation for a system designer, we suggest use of smaller 'radius' distribution in algorithm described in Section 3 for obtaining good algorithm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 1 Specifically, consider the self-expressiveness term in (3), Z Θe − Z Θe C 2 F . Since each data point z i (in the latent space) is approximated by a weighted linear combination of other points {z j } j=1,••• ,N (optionally, j = i) with weights c ij , this linear operation corresponds exactly to a set of linear neurons without non-linear activations.
Sentence 2: if we take each z i as a node in the network, we can then represent the self-expressiveness term with a fully-connected linear layer, which we call the self-expressive layer.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To parallel standard linear regression, {y i , x i } n i=1 are samples of (Y, X) linked through Y = X, β * + W . However, we assume each entry of x i is missing independently with probability ∈ (0, 1).
Sentence 2: the observed covariate vector x i takes the form x i,j = x i,j with probability 1 − * otherwise .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For simplicity, suppose this is the only strategy allowed by the set P. Then x t (ǫ) − E t−1 x t (ǫ) are independent increments when conditioned on the history.
Sentence 2: further, the increments do not depend on ǫ t . V T (P 1:T ) ≤ 2E T t=1 Y t where {Y t } is the corresponding random walk.
Relationship: reasoning