sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: For any x ∈ R + , the orbits in ( 5) correspond to a particular mechanical word 0, 1 or 0w1 depending on the value of x (Figure 1).
Sentence 2: for any word u, let y u be the fixed point of the mapping φ u on R + so that φ u (y u ) = y u and y u ∈ R + .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note also that choosing larger values ↵ leads to jumps in the solution sets sizes (c.f., 6.1).
Sentence 2: varying the required utility Q, even by a small amount, may not be possible to achieve by the current solution's size (↵ j ) and would require moving to a set larger by at least an ↵ factor (↵ j+1 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The decomposition into c-components forms a partition of the observable vertices, which gives very useful structural information about the causal model.
Sentence 2: a fact that is key to our whole analysis is that if N is a semi-Markovian Bayesian network on observable and unobservable variables V ∪ U with c-components C 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: 5 We evaluate this learning method on relatively simple instances on which commonly used software such as Alchemy can be a few orders of magnitude off from the optimal likelihood of the training data.
Sentence 2: table 3 compares the likelihood of the training data under the weights learned by FocusedFlatSAT and by Generative Weight Learning [7], as implemented in Alchemy, for four types of Markov Logic theories.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In the last section, we have shown that the symmetric NMF (2) can be transfered to problem (3), the latter admitting splitting property which enables us to design alternating-type algorithms to solve symmetric NMF.
Sentence 2: we exploit the splitting property by adopting the main idea in ANLS and HALS for nonsymmetric NMF to design fast algorithms for (3).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The set of payoffs reachable in Γ coincides with the set of payoffs reachable in Γ * .
Sentence 2: any strategy {λ σ } σ∈Σ1 , {ω σ } σ∈Σ1 over Γ * is payoff-equivalent to the realization-form strategy ω = σ∈Σ1 λ σ ω σ in Γ.
Relationship: entailment
Domain: NeurIPS
Sentence 1: As the exact choice of features, their encoding and comparison metric is not crucial to our technique, we chose a fairly simple representation that was general enough to use in a wide variety of settings, but informative enough to capture the details of objects (given the subtle variation that can distinguish two different cars, features such as [6] were found not to be precise enough for this task).
Sentence 2: we apply a first derivative Gaussian odd-symmetric filter to the patch at four orientations (horizontal, vertical, and two diagonal), giving four signed numbers per pixel.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The proposed embeddings should reflect the clinical relationships between procedures and diseases, such that the procedures are assigned to the diseases with short distance.
Sentence 2: for the m-th admission, we may recommend a list of procedures with length L, denoted as E m , based on its diseases and evaluate recommendation results based on the ground truth list of procedures, denoted as T m . given {E m , T m }, we calculate the top-L precision, recall and F1-score as follows: Pm+Rm .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Note that when the original graph G is a tree, the graph is already triangulated and the set T in Theorem 1 is equal to the edge set E. Hence, Theorem 1 implies that the inverse Γ of the augmented covariance matrix with sufficient statistics for all vertices and edges is graph-structured, and blocks of nonzeros in Γ correspond to edges in the graph.
Sentence 2: the (m − 1)p × (m − 1)p submatrix Γ V,V corresponding to sufficient statistics of vertices is block graph-structured; in the case when m = 2, the submatrix Γ V,V is simply the p × p block corresponding to the vector (X 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The remarkable performance came from the BSCI approach, which incorporates the convex LS formulation with the sparse RIR model.
Sentence 2: the BSCI approach yielded higher than 90% normalized correlation in source estimates when SNR was better than 20 dB, and it yielded higher than 99% normalized correlation in the low noise limit.
Relationship: entailment
Domain: NeurIPS
Sentence 1: More generally, given a matrix W and a pair of norms k that is, we apply the p-norm on the columns of W and the r-norm on the resulting d-dimensional vector.
Sentence 2: the 0 1 loss of a multiclass predictor h W on an example (x, y) is defined as the 0 1 loss equals 1 if h W (x) 6 = y and 0 otherwise.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We can now write the KKT conditions of problems (LRP λ ) and (LRD λ ) as In view of Eq.
Sentence 2: ( 3), we can see that if | θ * λ , xj < mλ, then the KKT conditions imply that the coefficient of xj in the solution β * λ is 0 and thus the j th feature can be safely removed from the optimization of (LRP λ ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The k-NN classifier is defined by We analyze the performance of g n,k by comparing it with g(x) = 1(η(x) ≥ 1/2), the omniscient Bayes-optimal classifier.
Sentence 2: we obtain bounds on Pr X (g n,k (X) = g(X)) that hold with high probability over the choice of data S, for any n. It is worth noting that convergence results for nearest neighbor have traditionally studied the excess risk R n,k − R * , where R n,k = Pr(Y = g n,k (X)).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We then possibly threshold β I1 with t 1 = 4λ n |I 1 | (to be specified), to obtain I 2 , repeat step 2 with I = I 2 to obtain β I and set all other coordinates to zero; return β.
Sentence 2: our algorithm is constructive in that it does not rely on the unknown parameters s, β min := min j∈S |β j | or those that characterize the incoherence conditions on X; instead, our choice of λ n and thresholding parameters only depends on σ, n, and p. In our experiments, we apply only the first two steps, which we refer to as a two-step procedure; the Gauss-Dantzig selector is a two-step procedure with the Dantzig selector as β init [5].
Relationship: entailment
Domain: NeurIPS
Sentence 1: Let V = {1, 2, . . . , n} denote the index set of the data points x i and partition V into c disjoint subsets V j ; i.e., V i ∩ V j = ∅ for i = j and ∪ c j=1 V j = V , where the cardinality of V j is n j so that c j=1 n j = n. We also make use of a matrix representation for the problem in question.
Sentence 2: we let X = [x 1 , .
Relationship: entailment
Domain: NeurIPS
Sentence 1: Theorem 2 states that a conditional probability on a countably-dimensional product space is uniquely defined (up to a.e.-equivalence) by a conditionally projective family of marginals.
Sentence 2: if we can define a parametric model on each finite-dimensional space Ω I x for I ∈ F(E) such that these models are conditionally projective, the models determine an infinite-dimensional parametric model (a "nonparametric" model) on the overall space Ω E x . Theorem 2 (Extension of Conditional Probabilities).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Using a mixed loss function, which is a weighted combination of the cross-entropy (XE) loss L XE and the reinforcement learning (RL) loss L RL , helps improve readability and fluency since the cross-entropy loss is based on a conditioned language model, with a goal of producing fluent captions.
Sentence 2: the mixed loss function is given by where γ ∈ [0, 1].
Relationship: entailment
Domain: NeurIPS
Sentence 1: For all S ⊂ S let q S and r S be the vectors of the probabilities of (a, b) and (c, d) respectively given that S is the set of parents that are active.
Sentence 2: then: this means that if there is only one parent shared between (a, b) and (c, d), M is the sum of two rank 1 matrices, and thus is at most rank 2.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Thus, for the above result to be non-trivial, we need 3.54λ ≤ 2, namely λ ≤ 0.56.
Sentence 2: in the noiseless case, the RoLR is able to estimate the LR parameter with a non-trivial error bound (also known as a "breakdown point") with up to 0.56/1.56 × 100% = 36% of the samples being outliers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: For a node w, father (w) denotes its parent, T(w) denotes the subtree tree rooted at the node, Ivs(w) denotes the number of leaves in the subtree and path( w) := w is the path from the root to the node.
Sentence 2: we use the path w from root to node as the label of the node w. We denote by words(S(x )) the set of all strings w such that wu E nodes(S(x )) for some (possibly empty) string u, which means that words(S(x)) is the set of all possible substrings of x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: When data are corrupted by noise of small magnitude such as Gaussian noise, a straightforward extension of SSC is the Lasso type method called "Lasso-SSC" [18,13]  Note that while Formulation (2) has the same form as Lasso, it is used to solve the subspace clustering task.
Sentence 2: the support recovery analysis of Lasso does not extend to this case, as X −i typically does not satisfy the RIP condition [18].
Relationship: entailment
Domain: NeurIPS
Sentence 1: This algorithm is not only highly efficient in practice, but also enjoys strong computational and statistical guarantees in theory.
Sentence 2: by leveraging a sophisticated characterization of local restricted strong convexity and Hessian smoothness, we prove that within each stage of convex relaxation, our proposed algorithm maintains the solution sparsity, and achieves a (local) quadratic convergence, which is a significant improvement over (local) linear convergence of proximal gradient algorithm in [9] (See more details in later sections).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We conclude this section by noting that the solvable case (29) is generic: its key results extend to the general situation defined above (21).
Sentence 2: we checked this fact numerically for several values of L. the minimization of F ∞ nullifies as many trial parameters as necessary to express the remaining parameters via independent effective parameters t 0 , t 1 , . . ..
Relationship: entailment
Domain: NeurIPS
Sentence 1: The resolution is to simply note that there is so little probability mass in those far reaches that the cost there is irrelevant.
Sentence 2: consider a single center p (and assume x → x − p 2 2 is integrable); the dominated convergence theorem grants where a ball B i may be chosen so that B c i x − p 2 2 dρ(x) ≤ 1/1024.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Condition (A) d T k Hd k ≤ 0 (at Step 3) may hold when the local model is not strictly convex (or H is not positive definite).
Sentence 2: d k is a direction of zero or negative curvature; a typical exploitation of non-positive curvature is to set ∆" equal to the "step to the trust-region boundary along that curvature segment (in Step 3)" as a model minimizer in the trust region.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We introduce the notation Q(Z, α) to denote the quantile of order 1 − α for the distribution of a random variable Z conditioned on the event Y = −1.
Sentence 2: the following quantile will be of interest: where we have used here the notion of generalized inverse F −1 of a càdlàg function F : We now turn to the definition of the ROC curve.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Then, there exists a sufficiently large k 0 ∈ N such that for all k ≥ k 0 the sequence {f (x k )} k generated by CR satisfies From Theorem 2, it can be seen that the function value sequence generated by CR has diverse asymptotic convergence rates in different regimes of the KŁ parameter θ.
Sentence 2: a larger θ implies a sharper local geometry that further facilitates the convergence.
Relationship: entailment
Domain: NeurIPS
Sentence 1: The vast majority of supervised learning techniques assume that source (training) data and target (testing) data are drawn from the same distribution over pairs of example inputs and labels, P (x, y), from which the conditional label distribution, P (y|x), is estimated as P (y|x).
Sentence 2: data is assumed to be independent and identically distributed (IID).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Fix 2 (0, 1) and L 2 N. A learner strategy, , is ( , L)-private if, for any adversary estimator, b X a , where the probability is measured with respect to the randomness in the target, X ⇤ , and any randomness employed by the learner strategy and the adversary estimator.
Sentence 2: 2 if a learner employs a ( , L)-private strategy, then no adversary estimator can be close to the target within an absolute error of /2 with a probability great than 1/L.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To tackle this limitation of conventional attention mechanisms, we propose to allow the attention model to output uncertainty on each feature (or input) and further leverage them when making final predictions.
Sentence 2: we model the attention weights as Gaussian distribution with inputdependent noise, such that the model generates attentions with small variance when it is confident about the contribution of the given features, and allocates noisy attentions with large variance to uncertain features, for each input.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Our solution to obtain more robust policies from few demonstrations, which are also capable of diverse behaviors, is to build on GAIL.
Sentence 2: to enable GAIL to produce diverse solutions, we condition the discriminator on the embeddings generated by the VAE encoder and integrate out the GAIL objective with respect to the variational posterior q (z|x 1:T ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: The stronger notion, node privacy, corresponds more closely to what was achieved in the case of traditional data sets, and to what one would want to protect an individual's data: it ensures that no matter what an analyst observing the released information knows ahead of time, she learns the same things about an individual Alice regardless of whether Alice's data are used or not.
Sentence 2: no assumptions are needed on the way the individuals' data are generated (they need not even be independent).
Relationship: entailment
Domain: NeurIPS
Sentence 1: At the bottom level, we have a set of distinct urns which are used to define the initial and transition probabilities of the HMDP model (and are therefore referred as HMM-urns).
Sentence 2: one of HMM urns, u 0 , is set aside to hold colored balls to be drawn at the onset of the HMM state-transition sequence.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Finally, it receives the edge semi-bandit feedback and uses it to update M t and B t . It is worth emphasizing that IMLinUCB is computationally efficient as long as ORACLE is computationally efficient.
Sentence 2: at each round t, the computational complexities of both Step 1 and 3 of IMLinUCB are O |E|d 2 .
Relationship: entailment
Domain: NeurIPS
Sentence 1: The observation model is defined to be outlier-prone of order n, if p(f |y 1 , ..., y n+1 ) → p(f |y 1 , ..., y n ) as y n+1 → ∞ [3,4].
Sentence 2: the effect of a single conflicting observation to the posterior becomes asymptotically negligible as the observation approaches infinity.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In mean-prediction, we consider the next timestep xt conditioned on xt−1 from the Gaussian prediction [8]: where k X (x) is a vector containing k X (x, x i ) in the i-th entry and x i is the i th training vector.
Sentence 2: we set the latent position at each time-step to be the most-likely (mean) point given the previous step: xt = µ X (x t−1 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: From the table, we observe that DBA achieves significantly better performance than all the other methods.
Sentence 2: for Accuracy@1 scores, DBA can achieve a gain of about 30% relative to two MNB methods, and about 10% relative to two SVM methods; for macro-average F-measures, DBA can achieve a gain of about 50% over MNB methods, and about 25% over SVM methods.
Relationship: entailment
Domain: NeurIPS
Sentence 1: [12] recently proved that the adversarial multi-armed bandit with switching costs is a hard online learning problem, which is a problem where the best possible regret guarantee is Θ(T 2/3 ).
Sentence 2: for any learning algorithm, there exists an oblivious setting of the loss values that forces a regret of Ω(T 2/3 ).
Relationship: entailment
Domain: NeurIPS
Sentence 1: Even when the estimation does not achieve the best performance, the metric always reduces the MSE from the original NW estimation.
Sentence 2: in the Slice, Pol, CPU, NavalC, and NavalT datasets, GGL performs poorly showing the non-Gaussianity of data, while the metric using the same information effectively reduces the MSE Figure 3: Regression with real-world datasets.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to produce an equivalent QP, however, we must also ensure that the dual variables µ x (y i , y j ), µ x (y i ) are the marginals resulting from a legal density α(y); that is, that they belong to the marginal polytope [4].
Sentence 2: we must enforce consistency between the pairwise and singleton marginals (and hence between overlapping pairwise marginals): yi µ x (y i , y j ) = µ x (y j ), ∀y j , ∀(i, j) ∈ E, ∀x.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Despite the difference in illumination and cell density from the training data, and despite the segmentation artifacts caused by the compression of the image sequence, our method shows a high generalization capability and obtains a total loss of 0.78%.
Sentence 2: we extract 93.2% of 384 mitosis events which is a significant improvement over the mitosis detection rate reported in [12] (81.5%, 294 events).
Relationship: entailment
Domain: NeurIPS
Sentence 1: We will assume throughout that all potential function values, all message values and all the required products computed by the algorithm can be represented as n-bit natural numbers and thus fall in Z N = {0, . . . , N − 1} where N = 2 n . As expressed by Equation (2), marginal probabilities are obtained by taking products of such n-bit numbers and then normalizing to obtain finite-precision real-valued numbers in the range [0, 1].
Sentence 2: it will be convenient to think of values in Z N as elements of the cyclic group of order N with addition and subtraction modulo N . we will make frequent use of the following simple fact: for any fixed x ∈ Z N , if r ∈ Z N is chosen randomly among all n-bit numbers, then x + r mod N is also distributed randomly among all n-bit numbers.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We numerically estimate the growth of regret in L, the exponent of L, in the log-log space of L and regret.
Sentence 2: since log(f (L)) = p log(L) + log(c) for any f (L) = cL p and c > 0, both p and log(c) can be estimated by linear regression in the new space.
Relationship: entailment
Domain: NeurIPS
Sentence 1: To capture this intuition, we add an additional term that aims to maximize the information gained about the answer from looking at the image.
Sentence 2: we introduce another adversarial regularizer corresponding to the difference in entropies between the base model prediction given the image and the question-only model which we write as We note that this regularizer resembles the conditional mutual information (CMI) between the answer and image given the question I(A; I|Q); however, f Q (q) is not constrained to be the marginal of f (v, q) such that estimating the CMI in this way is ill-defined.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In order to address the second drawback, we note that our aim at an iteration t of the algorithm is to reduce the potential function Φ(y).
Sentence 2: given the current distribution parameterized by θ Mt we would like to add a new tree θ Tt to the mixture that solves the following problem: θ T ∈ T . Here, T is the set of all tree distributions defined over n random variables.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In fact, in Section 3 we see how a rate of 1/n on the objective corresponds to a rate of 1/ √ n on the loss.
Sentence 2: we perform an oracle analysis of the optimum of the SVM objective (rather than of empirical minimization subject to a norm constraint, as in other oracle analyses of regularized linear learning), based on the existence of some (unknown) low-norm, low-error predictor w. Strong convexity is a concept that depends on a choice of norm.
Relationship: entailment
Domain: NeurIPS
Sentence 1: Instead, we formulate the estimation equations in terms of the components of the SVDs of the training data matrices.
Sentence 2: we take the SVDs of the zero-mean input and output matrices: From these two decompositions, we can write the two correlation matrices as Now we can rewrite the equation for A x to remove the need for the squaring operation and similarly for Ay  Using these identities, we compute A x and Ay using the following steps: 1) Find the SVDs of the data matrices using the expressions in equation 10.
Relationship: entailment
Domain: NeurIPS
Sentence 1: If the subspaces are disjoint or independent 1 , the solution X in both formulations is shown to be such that X ik = 0 only if data points y i and y k belong to the same subspace [7,14,15,25].
Sentence 2: the sparsest/lowest rank solution is obtained when each point y i is represented as a linear combination of points in its own subspace.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We used a perturbation size large enough to be noticeable by humans on the computer screen but small with respect to the image intensity scale (✏ = 32; see Figure 1a middle).
Sentence 2: we chose ✏ to be large (to improve the chances of adversarial examples transfer to time-limited human) but kept it small enough that the perturbations are class-preserving (as judged by a no-limit human).
Relationship: entailment
Domain: NeurIPS
Sentence 1: To do this, we parameterize the set of bias fields using the sine/cosine basis images shown on the right of Figure 1: We optimize Equation ( 8) by simultaneously updating the bias field estimates (taking a step along the numerical gradient) for each image to reduce the overall entropy.
Sentence 2: at time step t, the coefficients α j for each bias field are updated using the latent image estimates and entropy estimates from time step t − 1.
Relationship: entailment
Domain: NeurIPS
Sentence 1: We henceforth refer to this case as the state aggregation case.
Sentence 2: we assume that for any t = 0, 1  Note that dim E [Q] = K. We define the distance between Q ⇤ and the hypothesis class Q as The following result establishes that with Q and ⇢ defined above, the performance loss of OCP is larger than 2⇢H(H + 1) in at most K episodes.
Relationship: entailment
Domain: NeurIPS
Sentence 1: In a 3-clique, there is only one way to do this, so M is unique if G is triangulated.
Sentence 2: there exists a surjection from perfect matchings in the expanded dual of G to cuts in G. Furthermore, since we have given edges interior to the cliques of the expanded dual zero energy, every perfect matching M complementary to a cut C of our Ising model (2) obeys the relation This means that instead of a minimum-weight cut in a graph we can look for a maximum-weight perfect matching in its expanded dual.
Relationship: entailment
Domain: NeurIPS
Sentence 1: It turns out the dependency of F on X * can be essentially decoupled by using degenerated kernels corresponding to a finite number of feature maps.
Sentence 2: we consider kernels of form where we assume the number m of features is no larger than the particle size n. Then, the fixed point of SVGD reduces to where the test function f (x) := P x φ (x) no longer depends on the fixed point X * .
Relationship: entailment