sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Then, if γ is fully known, p γ (z|a obs , u) is computed with an extension to the Kalman smoothing algorithm to sequences with missing data, after which samples from p γ (a un |z) could be readily drawn.
Sentence 2: while the focus in this work is to define a generative model for high dimensional videos of simple physical systems, several recent works have combined physical models of the world with deep learning to learn the dynamics of objects in more complex but low-dimensional environments [3,4,9,34].
Relationship: neutral
Domain: NeurIPS
Sentence 1: The first row of Figure 3 contains similar plots that allow comparison of the four models we considered.
Sentence 2: figure 5 shows that the mean rankings of participants in Cluster 1 (N = 8) were best fit by the absolute utility model, the mean rankings of participants in Cluster 2 (N = 12) were best fit by the relative utility model, and the mean rankings of participants in Clusters 3 (N = 3) and 4 (N = 6) were better fit by feature-based models than by either the absolute or relative utility models.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The common choice is principal component analysis (PCA) feature, which is able to reduce the data dimension and identify the most important features [1].
Sentence 2: since the differences between the pixels of the unrelated "baboon" image or "tiger" image and the pixels of the images from two datasets are relatively small, the contiguous occlusion caused by these unrelated images is much more challenging than by random black or white dots.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Thus a linear PSR model is specified by Q and by the weight vectors in the above equation m aoq for all a ∈ A, o ∈ O, q ∈ Q ∪ φ (where φ is the null sequence).
Sentence 2: this is the first nonlinear PSR that can be used to represent a general class of problems.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Motivated by statistical modeling of biological sequences, Apostolico and Bejerano [1] showed that the bound on the maximal depth can be removed by devising a smart modification of Ron et.
Sentence 2: in this paper we describe a variant of prediction trees for which we are able to devise a learning algorithm that grows bounded-depth trees, while remaining competitive with any fixed prediction tree chosen in hindsight.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A training procedure based on pointwise log likelihood, for example, would prefer to reduce the posterior probability for a correct label from 0.6 to 0.4 in return for improving the posterior probability for a hopelessly incorrect label from 0.0001 to 0.01.
Sentence 2: unlike Och's algorithm, our method does not require one to provide a small set of candidate parses, instead relying on efficient dynamic programming recurrences for all computations.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Thus, by choosing η = O(1/ √ T ), OGD achieves an O( √ T (1 + P T )) dynamic regret, that is universal.
Sentence 2: in Step 3 of Algorithm 2, each expert submits its prediction x η t to the meta-algorithm, and receives the gradient ∇f t (x Observe the loss function f t (•) 7: Update the weight of each expert by 8: Send gradient ∇f t (x η t ) to each expert E η 9: end for Algorithm 2 Ader: Expert-algorithm Require: The step size η 1: Let x η 1 be any point in X 2: for t = 1, . . . , T do 3: Submit x η t to the meta-algorithm 4: Receive gradient ∇f t (x η t ) from the meta-algorithm 5: x η t+1 = Π X x η t − η∇f t (x η t ) 6: end for to get the prediction for the next round.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this case, the number of information sets |I 1 | = |S|H, and so polynomial in both states and the horizon.
Sentence 2: we take the Bayesian approach to uncertainty where the true MDP parameters are assumed to be distributed according to a known distribution, i.e., the true MDP M * is distributed according to an arbitrary distribution P(M).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The proof above uses our assumption that the clustering function f is defined on the set of all distance functions on n points.
Sentence 2: we then define a distance function to be any function d : S × S → R such that for distinct i, j ∈ S, we have d(i, j) ≥ 0, d(i, j) = 0 if and only if i = j, and d(i, j) = d(j, i).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We compare our method with CRF and SSVM models.
Sentence 2: let p i be a vector with length k, where its a-th element contains P (ŷ i = a|x), and let Q i,j be a k-by-k matrix with its (a, b)-th cells store P (y i = a, yj = b|x).
Relationship: neutral
Domain: NeurIPS
Sentence 1: A naive simple Monte Carlo estimator of the KL divergence between the output distributions of two inference algorithms requires evaluating output probabilities for both algorithms.
Sentence 2: second, BREAD estimates divergences between the target algorithm's sampling distribution and the posterior distribution, but the exact posterior samples necessary for BREAD's theoretical properties are only readily available when the observations y that define the inference problem are simulated from the generative model.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We considered the objective functions ¢ !'
Sentence 2: to ensure G BC & e we distribute weight down the tree from level g (the root, C ) to levels e ¢ &% H¢ ¥¤ ¦¤ ¦¤ by a convergent series; where for simplicity we set the total weight allocated at level ' , G # C , to be G # C ¡ § # )( ¡ § # )( 10 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The Maximum a Posteriori (MAP) estimation of network parameters W given training data {X , Y} for learning multiple tasks is where we assume that for prior p (W), the parameter tensor of each layer W is independent on the parameter tensors of the other layers W = , which is a common assumption made by most feedforward neural network methods [3].
Sentence 2: the above updating equations ( 11) are computationally prohibitive, due to the dimension explosion of the Kronecker product, e.g.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that L-KM uses much less memory compared to OSM, converges at almost the same rate in iterations, and its running time per iteration improves as the iteration count increases.
Sentence 2: subsequent work on the proximal setup uses trust regions, penalty functions, level sets, and other more complex algorithmic tools; we refer the reader to [18] for a survey on bundle methods.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Further for the Pascal dataset we do not have any initial depth estimate.
Sentence 2: as shown, our joint approach achieves an improvement of almost 2.3% , and 1.2% in the overall accuracy and average intersection-union (I/U) score over the model of AHCRF [9], and almost 1.5 % improvement in the Algorithm Av.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (F2) is more stringent and we show that it is satisfied for convex nondecreasing and concave nonincreasing functions.
Sentence 2: . . , w m ) ∈ R m be a random vector with independent components w i which satisfy Ew i = µ and w i − µ ψ2 ≤ K. Let imp(w) := m 2 −1 1≤i<j≤m (w i − w j ) 2 be the empirical variance of w. Then, for u ≥ 0, We can immediately apply this result when k / ∈ S. for k ∈ S, a more careful application is needed since we can only guarantee an exchangeable distribution for y k [m] in this case.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We then have to solve the prediction problem, i.e.,  We now notice that if the scalar c j = c(j), where c is a non-increasing function of rank position j, then ( 17) can be solved simply by sorting the values of h i , ✓i in decreasing order.
Sentence 2: in the case of tractable graphical models, dynamic programming can be used to efficiently perform inference.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For example, if P 1 is close to one, L ≈ C2 p(x)dx, while P e ≈ C1 q(x)dx.
Sentence 2: in such an inner product space, the Cauchy-Schwarz inequality holds, that is, p, q 2 f ≤ p, p f q, q f . Based on this discussion, an information theoretic distance measure between the two pdfs can be expressed as We refer to this measure as the Laplacian pdf distance, for reasons that we discuss next.
Relationship: neutral
Domain: NeurIPS
Sentence 1: These studies show that frequency and AoA are highly correlated (typically r = -.6) explaining the confound of older studies on frequency.
Sentence 2: recently, Ellis & Lambon-Ralph (in press) have shown that when pattern presentation is staged, with one set of patterns initially trained, and a second set added into the training set later, strong AoA effects are found.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the simplified case discussed in Section 2.2, with linear kernel on X = R d and H = R k finite dimensional, we have F = R d and this assumption corresponds to require the existence of a matrix W * ∈ R k×d = H ⊗ F, such that g * (x) = W * x for any x ∈ X .
Sentence 2: as it is standard in statistical learning theory, we can impose regularity conditions on the learning problem, in order to derive also generalization bounds for f . if we denote by F the RKHS associated to the kernel k, we will require g * to belong to the same space H ⊗ F where the estimator g introduced in Eq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given fixed base distributions, we can optimise the KL bound with respect to the parameters A = LU and b.
Sentence 2: provided we choose a base distribution class that includes the Gaussian as a special case (for example generalised-normal, skew-normal and asymptotically Student's t) we are guaranteed to perform at least as well as classical multivariate Gaussian KL approximate inference.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, at high dimensions, the relative error made as considering E||X|| instead of the random variable ||X|| becomes asymptotically negligible [2].
Sentence 2: under this assumption correlation between W and noise W − Q(W ) = ( 0 , 1 , ..., N −1 ) is negligible, and can be approximated as an additive noise.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Proof: Expected Generative Solution is ML Solution In the case of a generative model which has no constraints (i.e., can model any distribution), the maximum likelihood solution will have distribution px(a) = .J y Ef':18(yi -a), i.e., the model will produce only the observations and all of them with equal probability.
Sentence 2: without Green's function, one might expect local optima in which clusters of points collapse onto a single location.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our ablation studies (Section 4.3) support the hypothesis that the increased performance is due to the uncertainty-dependent reweighting of targets, as well as demonstrate that the performance of STEVE consistently increases with longer horizon lengths, even in complex environments.
Sentence 2: we show that we can do still better by weighting the candidate targets in a way that balances errors in the learned Q-function and errors from longer model rollouts.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The goal of information bottleneck with side information (IBSI) is therefor to find a stochastic map of % to a new variable , @ 9¦ ¢¡ D © , in a way that maximizes its mutual information with & ' and minimizes the mutual information about & £) .
Sentence 2: is supported a grant from the ministry of Science, Israel.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the Newton-LASSO Method, we approximate the objective function F at the current iterate P k by the piecewise quadratic model ) where g k = L (P k ), and similarly for H k . The trial step of the algorithm is computed as a minimizer of this model, and a backtracking line search ensures that the new iterate lies in the positive definite cone and decreases the objective function F . We note that the minimization of q k is often called the LASSO problem [18] in the case when the unknown is a vector.
Sentence 2: the NL-FISTA algorithm terminated the FISTA iteration when the minimum norm subgradient of the LASSO subproblem q F became less than 1/10 of the minimum norm subgradient of F at the previous step.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We select the criticism points as the maximizers of this cost function: Where [n]\S denote all indexes which not include the prototypes, and c ⇤ is the number of criticism points desired.
Sentence 2: it has quickly developed into an important part of model design, and Bayesian statisticians now view model criticism as an important component in the cycle of model construction, inference and criticism (Gelman et al., 2014).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Moreover, as the codebook size grows, non-linear classifiers, represented by RBF SVM in our experiments, suffer from the curse of dimensionality partially due to the large dimensionality of the over-complete feature representation.
Sentence 2: we propose a new stacking technique for building a deep architecture, using a linear SVM as the base building block, and a random projection as its core stacking element.
Relationship: neutral
Domain: NeurIPS
Sentence 1: D-TS + : Implement the same operations as D-TS, except for the selection of the first candidate (Line 11 in Algorithm 1) is replaced by the following two steps: a (1) ← arg min R( 1) i ; D-TS + only changes the tie-breaking criterion in selecting the first candidate.
Sentence 2: corollary 1 provides an upper bound for the regret of D-TS + .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Assuming the adjacencies X -Y -Z -W have been found, conditional-independence tests will reveal that (X Jl Z) and (X .,il Z I Y), which is a sufficient condition for the V-structure X ---> Y f -Z.
Sentence 2: it contains spurious links to its spouses, to members of its district, to members of its children's district, and to parents of nodes in those districts, which we all call SD links (for Spouse/District).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The reason why these factors are less than the 3000 in the previous setup is that, for the larger proteins, the number of tied states is typically much higher than that for the small ones.
Sentence 2: there always exists a maximizing µ that is integral and corresponds to x M . Although the number of variables in this LP is only O(|E| + |V |), the difficulty comes from an exponential number of linear inequalities typically required to describe the marginal polytope M(G).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Another short-coming of the Ising model is that it can only model binarized data and cannot be normalized for integer y-s [6], so either the time bins need to be reduced to ensure no neuron fires more than one spike in a single bin or the spike counts must be capped at 1.
Sentence 2: comparison with the output distribution derived for the Gaussian RLM suggests that this choice would fail to capture the instantaneous covariance that the LDS formulation transfers to the output distribution (and which appears in the low-rank structure of S above).
Relationship: neutral
Domain: NeurIPS
Sentence 1: In fact, the use of ensembles of CART trees [4] not only inherits all positive aspects of gradient boosting (robustness, speed and insensitivity to hyper-parameters) but is also a natural match for metric learning.
Sentence 2: they often demand high computational expense, not only in parameter fitting but also in model selection and hyper-parameter tuning.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Consider ✓ defined as follows: ✓t = ✏, and ✓t = Using the fact that log(1 z) 2z for z 2 [0, 1 2 ], it follows that We have ✓t ✓ ⇤ t for t 6 = t, and so Recall from Lemma 3 that, for each word i and t = arg max t it ✓ ⇤ t , we have ✓ t > K(↵, T, N ).
Sentence 2: it is important to note that we only restrict the maximum number of topics per document, letting the Dirichlet prior and the likelihood guide the choice of the actual number of topics present.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Now, if parameter a in a,c is very close 6 to 1, then |Y * t | = K, and the algorithm itself will produce ordered subsets Ŷt such that | Ŷt | = K. Moreover, it does so by receiving full feedback on the relevant classes at time t (since Y t ∩ Ŷt = Y t ).
Sentence 2: the algorithm is not directly dealing with the original loss when making exploration.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This class of models strike a nice balance between their representation power (e.g., ability to model cliques) and the complexity of learning and inference processes on these structures (e.g., message passing is exact on trees).
Sentence 2: a commonly used kernel function, which we will focus on, is the Gaussian RBF kernel k(x, x For Gaussian RBF kernel, there exists a feature map φ : R → F such that k(x, x ) = φ(x), φ(x ) F , and the feature space has the reproducing property, i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The covariance between f x and f z is fully determined by the coordinates of the data pair x and z, and is defined by any Mercer kernel function K(x, z).
Sentence 2: we employ the linear kernel K(x, z) = x • z or the Gaussian kernel K(x, z) = exp − κ 2 x − z 2 2 , and shift the origin of the kernel space to the empirical mean, i.e.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We can then consider a quadratic regularizer of the form cV n /2kwk 2 to define the regularized empirical risk R n (w) := L n (w) + (cV n /2)kwk 2 and the corresponding optimal argument Since the regularization in ( 4) is of order V n and (3) holds, the difference between R n (w ⇤ n ) and L(w ⇤ ) is also of order V n -this may be not as immediate as it seems; see [22].
Sentence 2: in the main paper we only use the protein homology dataset provided on KDD cup 2004 website.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The inter-agent communication and the LP avoid the exponential blowup in the state and action spaces, having computational complexity dependent, instead, upon the induced tree width of the coordination graph used by the agents to negotiate their action selection.
Sentence 2: rather than placing a priori restrictions on the communication structure between agents, we first choose the form of an approximate value function and derive the optimal communication structure given the value function architecture.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Intuitively, if at least 100ρ% of all policies are feasible, L(v; ρ) is less than the objective value for the unconstrained CE method and greater than the expected G-value over the 100ρ% policies of the highest H-values.
Sentence 2: a policy π ∈ Π is an optimal policy in Π with respect to ] be the expected cost over trajectory distribution P π .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Less well-known is a generalization of this result to renewal processes [13].
Sentence 2: instead of using a single constant λ * , one could use (say) a piecewise linear function λ * (t) The more segments we use, the more flexibility we have; the price being the complexity of resampling this function, and slower mixing because of correlations it introduces.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Formally, a task τ ∈ T is defined as an MDP, and a policy π(s) is a function that outputs an appropriate action a given state s. The return, or utility, generated from interacting with the MDP environment by following π over an episode of k steps, is defined as the cumulative reward u = k i=1 r i , where r i is the immediate reward received by acting at step i − 1.
Sentence 2: so, detecting from only one single perspective of policy's performance is not enough to exactly reason about the opponent's policy.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Principally, we have been assuming that each individual signatory of the petition exposes her petition copy independently with probability δ.
Sentence 2: in this work, we have developed techniques that allow us to reason about the full tree along which the information spreads from the portion that is observed; as a consequence, we are able to propose estimates for the size of a network cascade from a sample of it, and to deduce certain structural properties of the tree that it produces.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We assume that each observation y i,j is a point on a N -dimensional Riemannian manifold M embedded in R P (with P ≥ N ) and equipped with a Riemannian metric g M . We denote ∇ M the covariant derivative.
Sentence 2: each data point consists in four normalized scores, which can be seen as a point on the manifold M =]0, 1[ 4 .
Relationship: neutral
Domain: NeurIPS
Sentence 1: There are three key results: 1.
Sentence 2: (a) RFs in inferotemporal cortex are indeed very large, allowing for the possibility of distractor interference in the low load condition; and (b) even under the attentional load hypothesis, the only reason that an unattenuated distractor stimulus would interfere with target processing is that there is something in common about them, since it is known that there is more to the effects of distractors than just competition at the stage of the actual responses (Driver, 2001).
Relationship: neutral
Domain: NeurIPS
Sentence 1: • If h✓ h0i , ✓ ⇤ i < 0, then for all t > T , Theorem 2 implies that if we use random initialization for ✓ h0i , with probability one, the Population EM 2 estimates converge to the true parameters.
Sentence 2: in the M-step of EM, when the mean of the estimated component Ĉ is updated, it is pulled rightward towards +1.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (2006) has shown that banded Toeplitz matrices asymptotically approach circulant matrices as the size of the matrix grows.
Sentence 2: since the distance from the origin during the initial random walk phase grows as √ n, the process will start to deviate from a random walk after n c ∼ (α(2 − α)) −1 steps.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Hence, the posterior distribution 𝑝(ℎ|𝑥) has the form 𝑝(ℎ|𝑥) = 𝑝(ℎ)𝑝(𝑥|ℎ) 𝑝 (𝑥) .
Sentence 2: there is a base function 𝑓 0 : R 𝑑 → R, centers 𝜇 1 , 𝜇 2 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The idea of much prior work on low-rank kernel approximation is to find K which is nearly as good as K k , but can be computed much more quickly.
Sentence 2: 2.1 Lower bound for low-rank approximation of MM T . We first illustrate our lower bound technique by showing hardness of direct approximation of MM T . Theorem 3 (Hardness of low-rank approximation for MM T ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: This would make it easier for arbitrary regions of space to share the same covariance structures; at present the areas 'controlled' by a particular expert tend to be local.
Sentence 2: in section 5 we present the results of several simple simulations that highlight some of the salient features of our proposal, and finally in section 6, we discuss our work and place it in relation to similar techniques.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Furthermore, to evaluate the computational speedup gained by using random features, we compare against Nyström method [Drineas and Mahoney, 2005] as a secondary baseline.
Sentence 2: the spectrum of C plays a crucial role in our results through the following key-quantity: Essentially, we will see that the constant κ(B k , k, m) is the dominating factor when bounding the excess risk, and, therefore, will determine the rate of convergence of our algorithms.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The nontenninals A, B, C are elements of the non-terminal alphabet Vnt , which includes the start symbol S. t is a string of tenninal symbols from the terminal alphabet Vt 1 • For determining the language L of a certain grammar G I use simple depth-first exhaustive search of the derivation tree.
Sentence 2: unlike in usual interpretations of this proof, these constraints are not strict (some grammars are better learnable than others, allowing for an infinite "Grammar Universe"), and they are not a-priori: they are the outcome of iterated learning.
Relationship: neutral