sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Figure 3a shows a boxplot of the L 2 loss of the predicted Y values with respect to the true values for both the baseline and our method, considering the 121 cases out of 200 in which our method does produce an answer.
Sentence 2: as the transfer bias can become arbitrarily large (as we have seen in Example 1), the prediction accuracy of this solution strategy may be arbitrarily bad (even in the infinite-sample limit).
Relationship: neutral
Domain: NeurIPS
Sentence 1: An immediate consequence is that given error tolerance 2 (0, 1) and with choice 2 (0, 1 4 2 ), then upper bound (10) is at most (by having each of the two terms on the right-hand side be  2 ) if n > m log 2m (i.e., = 1 + log 2 / log m), and This means that if we have access to a large enough pool of labeled time series, i.e., the pool has ⌦(m log m ) time series, then we can subsample n = ⇥(m log m ) of them to use as training data.
Sentence 2: as a running example used throughout this paper, consider a time series that tracks how much activity there is for a particular news topic on Twitter.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We define φ(s t , a) as a concatenation of the current feature vector and a meta-feature vector that provides information about previous classification results and cost.
Sentence 2: this method ignores the difference between distributions of states induced by executing the oracle's policy and the learner's, thus has a quadratic loss in the task horizon T . A recent approach called Dataset Aggregation [3] (DAgger) yields a loss linear in T by iteratively training the policy in states induced by all previously learned policies.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (2) If the Gram matrix is entry-wise non-negative and if the set of predictors indexed by {1, . . . , p} can be partitioned into subsets B 1 , .
Sentence 2: since Z j 2 ≤ X j 2 and e j Σ −1 SS X S / √ n 2 ≤ {φ min (S)} −1/2 for all j, the sub-Gaussian parameters of these collections are upper bounded by σ/ √ n and σ/({φ min (S)} 1/2 √ n), respectively.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The Markov matrix P defines an imposed diffusion process, shown by Coifman & Lafon (2006) to efficiently capture the diffusion geometry of a manifold M. The DM framework may be used for dimensionality reduction to embed data using the eigendecomposition of the diffusion operator.
Sentence 2: 3) 2: Define a sparsity measure ŝ(i), i = 1, ..., N (Eq.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 1 shows the segmentation for a typical image from which it can be seen that, moving horizontally from right to left, the only "legal" transitions in segmentation are B → F, F → O and O → B. Algorithm 1 is therefore modified to consider legal segmentations only.
Sentence 2: algorithm 1 gives pseudo-code for the incremental incorporation of a measurement and greedy labelling.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If the next hypothesis p t+1 of a ROME algorithm is on H t , then by Lemma 2, it and all other members of H t satisfy D(p t+1 ||u) = D(p t+1 ||p t ) + D(p t ||u).
Sentence 2: it improves on Pinsker's inequality (Lemma 1) when n = 2, p is small, and q is close to p. The following is a direct consequence of Lemma 8 and Theorem 6.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the setting of least-squares function approximation, the second-order algorithm of choice is commonly Gauss-Newton, which approximates the curvature as the Gauss-Newton matrix G := E[J T J], where J is the Jacobian of the mapping from parameters to outputs [18].
Sentence 2: the Euclidean norm of the change depends on the parameterization θ.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The rest of the paper is organized as follows.
Sentence 2: the second fundamental form II, which is an extrinsic quantity describing how the tangent spaces bend locally is, at every x ∈ M, a map II x : T x M → (T x M) ⊥ (in this case of class C 1 by Assumption 2) from the tangent space to its orthogonal complement (II(x) := B(x, x) in the notation of [16, p.
Relationship: neutral
Domain: NeurIPS
Sentence 1: ( §3) 3) In order to use attention mechanisms defined under our framework in an autodiff toolkit, two problems must be addressed: evaluating the attention itself and computing its Jacobian.
Sentence 2: we illustrate max Ω and Π Ω for various choices of Ω in Figure 2 (2-d) and in Appendix C.1 (3-d).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We partially labelled some of the images.
Sentence 2: to deal with the above we marginalize out S n and we sample z nk from the marginalized posterior conditional P (z nk |S −n , Z −(nk) , α, X) which is computed according to P (z nk |S −n , Z −(nk) , α, X) ∝ P (z nk |Z −(nk) , α) where P (z nk |Z −n,k , α) for the infinite case is computed as described in section 3.2 while computing the sum requires an approximation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this paper, we develop a novel approach to the problem of learning sparse representations in the context of fused sparsity and unknown noise level.
Sentence 2: the most appealing penalty, the number of nonzero elements of Mβ, leads to a nonconvex optimization problem which cannot be efficiently solved even for moderately large values of p. Instead, convex penalties of the form minimizer of the penalized log-likelihood To ensure the scale equivariance, the weights ω j should be chosen inversely proportionally to σ: ω j = σ −1 ωj .
Relationship: neutral
Domain: NeurIPS
Sentence 1: While potentially effective in practice, homotopy continuation methods require both a trade-off parameter for g(x; θ) and a pre-defined schedule or heuristic for adjusting θ, both of which could potentially be image dependent.
Sentence 2: noise and kernel dependencies notwithstanding, this reparameterization places the proposed cost function in a form exactly consistent with nearly all prototypical sparse regression problems, where 2 column normalization is ubiquitous, at least in part, to avoid favoring one column over another during the estimation process (which can potentially bias the solution).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Nodes C and D, children of a neutral "tax" topic, reveal how parties frame taxes as gains in terms of new social services (Democrats) and losses for job creators (Republicans).
Sentence 2: after choosing the table (indexed by t d,s ), if the table is new, the group will select a combo of dishes (i.e., a path, indexed by c d,t ) from the tree menu.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Assume condition (Kernels-B) holds.
Sentence 2: if several test samples are to be treated in succession, and we want to avoid a full, resource-consuming re-training using all the training samples each time, an interesting alternative is the following: learn once a function f 0 (P X , x) using the available training samples via (3); then, given a partially labeled test sample, learn a decision function on this sample only via the usual kernel norm regularized empirical loss minimization method, but replace the usual regularizer term f 2 by f − f 0 (P x , .)
Relationship: neutral
Domain: NeurIPS
Sentence 1: 4 Furthermore, note that 1(x 0 ) (I − {P (x |x, π B )} x,x ∈X ) −1 1(x) represents the total visiting probability E[ from the initial state x 0 to any state x ∈ X , which is a non-negative quantity.
Sentence 2: we include runs using discrete observations (a one-hot encoding of the agent's position) and image observations (showing the entire RGB 2D map of the world).
Relationship: neutral
Domain: NeurIPS
Sentence 1: -Convex minimization: A = ∂g and B i = ∂f i for a strongly-convex function g and smooth functions f i : we recover proximal-SVRG [24] and SAGA [3], to minimize min z∈E g(z) + i∈I f i (z).
Sentence 2: there is no need to assume the existence of a function We will also consider splits which are adapted to the saddle-point nature of the problem, that is, of the form B(x, y) = k∈K B x k (x, y), j∈J B y j (x, y) , which is a subcase of the above with I = J × K, B jk (x, y) = (p j B x k (x, y), q k B y j (x, y)), for p and q sequences that sum to one.
Relationship: neutral
Domain: NeurIPS
Sentence 1: To understand these optimality criteria intuitively, it is helpful to see their equivalence of the Gaussian distribution in the limit of large κ 1 , κ 2 and κ s . Under the condition x 1 ≈ x 2 , Eq.
Sentence 2: κ s determines the correlation between two stimuli, i.e., how informative one cue is about the other, and it regulates the extent to which two cues should be integrated.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For 2D datasets, we use the recently released Pix3D dataset to obtain 1, 515 RGB images of chairs alongside with their silhouettes [Sun et al., 2018a], with an addition of 448 clean background images crawled from Google image search.
Sentence 2: the discriminator D shape contains 6 volumetric convolutional layers and the generator G shape contains 6 strided-convolutional layers.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A sampling result then shows that a small random sample of these output trees can attain a significant fraction of the margin obtained by the complete graph.
Sentence 2: given any unit L 2 norm predictor w on the complete graph G, our task is to investigate how the margins Γ(w, x, y), for each (x, y) ∈ X ×Y, will be modified if we approximate the (true) expectation over all spanning trees by an average over the sample T . For this task, we consider any (x, y) and any w of unit L 2 norm.
Relationship: neutral
Domain: NeurIPS
Sentence 1: (2) The use of unlabeled data enhances our ability to avoid parameter over-fitting and over-estimation in grid based random fields even when using a learner that uses only approximate inference methods.
Sentence 2: assume we have a set of independent labeled images, D l = " (x (1) , y (1) )), , and a set of independent unlabeled images, " .
Relationship: neutral
Domain: NeurIPS
Sentence 1: Randomized neural networks are immortalized in this AI Koan: In the days when Sussman was a novice, Minsky once came to him as he sat hacking at the PDP-6.
Sentence 2: since others have already empirically demonstrated the benefits of random featurization [2,3,4,5], we only a present a few illustrations in this section.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Actually, they are random samples of a centered Gaussian process indexed by the space X with covariance structure given by 1 P ϕ(x), ϕ(x ) , where we use the notation u, v = i u i v i for two square-summable sequences u and v. Indeed, E A p [ψ p (x)] = 0, and The continuity of the initial features (ϕ i ) guarantees that there exists a continuous version of the process ψ p which is thus a Gaussian process.
Sentence 2: this is no more the case when P is not the Lebesgue measure and it seems difficult to modify the features ϕ i in order to recover the same bound, even when P is known.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For the general convex and Lipschitz continuous optimization problem, the iteration complexity of gradient (subgradient) descent is O( 1 ǫ 2 ), which is optimal up to constant factors [15].
Sentence 2: the objective function can be rewritten as (10) And the mixed gradient can be rewritten as gk = g + ∇g t (w t ).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Here, we introduce a new approach using a near-maximum entropy model, that makes this type of analysis feasible for very high-dimensional data-the model parameters can be derived in closed form and sampling is easy.
Sentence 2: while for small image patches the DG model provided an excellent fit to the true distribution, we were able to show that this agreement breakes down in the case of larger image patches.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our algorithm learned a good policy under significant observation noise in less than 30 seconds of system interaction.
Sentence 2: the joint distribution over the updated (random) belief and the (non-random) action is Next, the filtering 'prediction step' computes the predictive-distribution of b t+1|t = p(x t+1 |z 1:t , u 1:t ) from the output of dynamics model f given random input bt|t .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The constant c controls the relative importance of these two competing terms and it can be chosen via cross validation.
Sentence 2: given n input examples x 1 , ..., x n in ℜ D and corresponding class labels y 1 , ..., y n , the technique in [9] learns the D × D transformation matrix L that optimizes the following objective function: where η ij ∈ {0, 1} is a binary variable indicating whether example x j is one the k-closest points of x i that share the same label y i , c is a positive constant, y il ∈ {0, 1} is 1 iff (y i = y l ), and h(s) = max(s, 0) is the hinge function.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A different label is fed into DPN to switch among response policies by linking the shared convolution layer with the corresponding control layer, achieving convenient fast policy switch.
Sentence 2: deep BPR+ inherits all the advantages of BPR+ and empirically shows better performance in terms of detection accuracy, cumulative rewards and speed of convergence compared to existing algorithms in complex Markov games with raw visual inputs.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the experiments, we demonstrate that the markers obtained using our approach are capable of retaining bit strings that are long enough to be practical.
Sentence 2: one can set the recognizer to have a certain architecture, fix the variability and the strength of the adversarial effects that need to be handled, and then the synthesizer will adapt so that the most "legible" codes for such circumstances can be computed.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since Hodgkin and Huxley [1] described the ionic flux across the neuronal membrane with four nonlinear differential equations more than half a century ago, continuous efforts have been made either to extract an essence of the nonlinear dynamical aspect by simplifying the model, or construct ever more realistic models by including more ionic channels in the model.
Sentence 2: the leaky integrate-and-fire model [4], originally proposed far in advance of the Hodgkin-Huxley model, consists of only one variable that corresponds to the membrane potential, with a voltage resetting mechanism.
Relationship: neutral
Domain: NeurIPS
Sentence 1: aT (cp(X) -cp(y)) = 1, (20) where, as in ( 12), the optimal value of b will be given by b* = a; cp(x) -"'*Jar~cp(x)a* = a; cp(y) + "'*Jar~cp(y)a*, where a* and "'* are the optimal values of a and '" respectively.
Sentence 2: we can also write (23) as -- which is a second order cone program (SOCP) [5] that has the same form as the SOCP in ( 16) and can thus be solved in a similar way.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The only two free parameters are κ prod and κ r . Since only κ r has an direct effect on all sets of belief species, we fixed κ prod = 50 and varied κ r . Small values of κ r results in better approximation as less of the probability mass in each belief species set is in an unassigned state.
Sentence 2: together, reactions for recycling probability mass, implementing sum-product messages, and implementing product messages define a reaction network whose equilibrium computes the messages and marginal probabilities via the sum-product algorithm.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Finally, it can be shown that the CCA problem given in Problem ( 1) is ill-posed if the population auto-covariance matrices This observation follows from the fact that if there exists a direction in the kernel of in which x and y exhibit non-zero covariance, then the objective of Problem (1) is unbounded.
Sentence 2: the objective is no longer an expectation, but is instead a ratio of expectations.
Relationship: neutral
Domain: NeurIPS
Sentence 1: For example, if a strong artifact is present in only one of the data sets, the CSP is trained on the artifact and not on the differences caused by motor imagery [7].
Sentence 2: optimization of the center and extent of the ROI, either by a-priori knowledge gained by fMRI scans or by numerical optimization of the ERD in specific frequency bands, is expected to lead to higher SNRs and hence higher classification accuracy.
Relationship: neutral
Domain: NeurIPS
Sentence 1: If X has rank 1 then X 0 ≥ xx H 0 .
Sentence 2: this conversion is done by noting that for n × n Hermitian matrices A, B: So if we define the vector A v as an n 2 vector such that its elements are for this subproblem, the memory usage scales linearly with N , the number of measurements, and quadratically with n, the dimension of the data.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The Leaky Integration Unit architecture [22] proposed residual connections for RNNs but were unable to learn the state transition matrix due to the problem of exploding and vanishing gradients.
Sentence 2: fastRNN updates the hidden state h t as follows: where 0 ≤ α, β ≤ 1 are trainable weights that are parameterized by the sigmoid function.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A recent empirical study [6] of a large number of 3 × 3 images, represented as points in R 9 revealed that they approximately lie on a two dimensional manifold known as the Klein bottle.
Sentence 2: . . , Rx u } is at least 2 u−1 using the Johnson-Lindenstrauss Lemma [11] and the fact that {x 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: When performing PCA in practice with massive d and a large/growing dataset, an incremental method like that of Krasulina or Oja remains practically viable, even as quadratic-time and -memory algorithms become increasingly impractical.
Sentence 2: let Ω denote the sample space of all realizations (v no , x no+1 , x no+2 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The weighted size w 1,2 with respect to C 1 , C 2 , n 1 and n 2 is defined as Theorem 1.
Sentence 2: given a column centered data matrix A, set B = D(A) and Then, X is the optimal solution to Problem (1) iff we can determine whether X is the membership matrix of A by solving Problem (5).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Given a continuous model ¤ §¦ ¡ ©¨, we consider the following prediction rule: predict # ¥ if ¤ §¦ ¡ ¨ , and predict # $ ¥ otherwise.
Sentence 2: such an analysis would have led to a result of the following form for our problems: It is also interesting to compare Theorem 3.1 with the leave-one-out analysis in [7].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Intuitively, U-Net-1 to U-Net-k all have the same input size, while U-Net-1 views the smallest region with the highest resolution and U-Net-k views the largest region with the lowest resolution.
Sentence 2: we manually labeled the first 16 slices in one stack as the training data and used the other four stacks, each containing 81 sections, for evaluation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: First, we formulate our objective in terms of how to minimize an expected risk function with respect to the labeled training data L = L s ∪ L t and the translator φ by extending the risk minimization framework in [5].
Sentence 2: for data preprocessing, the SIFT descriptor [6] was used to find and describe the interesting points in the images, and then clustered the extracted interest points into 800 clusters to obtain the codebook.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In all our tests we centered and normalized the data, included a bias term (a linear intercept), and choose the regularization parameter as = 1/m, where m is the number of data points.
Sentence 2: the above is a sketch-and-project update for a linear system in R n 2 , which allows to obtain an alternative proof of Theorem 5, without using our results from Euclidean spaces.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We assume that the distributions of the mixture components h r differ only in a small number of dimensions, 1, . . . , d race and are identical on the other d other := d − d race dimensions.
Sentence 2: efficient coding is necessary given that the brain accounts for   [12], with the middle face being a 50% blend of the Asian and Caucasian faces on either side.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In a typical setting, we are given as input a protein interaction data set, specifying a set of proteins P and a set of observed interacting pairs T.O.
Sentence 2: we then use the EM algorithm [15], to fill in the details of the model, learning both the motif affinities and activities from the observed data of protein-protein interactions and protein motif occurrences.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This is coherent with our theoretical analysis, since GFM needs to estimate more parameters.
Sentence 2: additionally, one can easily construct families of probability distributions that obtain a relatively fast convergence rate as a function of m. Solving (2) becomes straightforward in the case of a specific distribution in which the probability mass is distributed over vectors y containing only a single positive label, i.e., m i=1 y i = 1, corresponding to the multinomial distribution.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Different with NAO that tries to reduce the huge computational cost brought by the search algorithm, weight sharing aims to ease the huge complexity brought by massive child models via the one-shot model setup [5].
Sentence 2: for the ease of introduction, we use the same notation x = {x 1 , • • • , x T } to denote such string sequence of an architecture x, where x t is the token at t-th position and all architectures x ∈ X share the same sequence length denoted as T . T is determined via the number of nodes B in each cell in our experiments.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The latter motivation can be "lifted" to the matrix domain by adding n virtual outcomes at I/n: The worst-case regret of classical Laplace after T iterations equals log T +n−1 n−1 ≤ (n−1) log(T +1) (see e.g.
Sentence 2: the worst-case regrets of classical and matrix LSM coincide.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Unfortunately because of extinct species, we don't have a perfect input and must account for noise.
Sentence 2: in general there is no reason to believe that PCA+ would perform well in clustering truly hierarchical data: there are regimes of the HSBM for which applying only phase 1 of the algorithm might lead to a high missclassification error and high cost and for which we can prove that LINKAGE++ is an p1`εq-approximation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Some guidance is offered in [6] regarding how c might be chosen.
Sentence 2: if we remove the slack function by setting s 2 to zero and let s 1 = −(1 − α)c T Φr, our LP (1) becomes maximize c T Φr (2) subject to min u (g u + αP u Φr) − Φr ≥ 0, which is precisely the LP considered in [6] for approximating the optimal cost-to-go function in a discounted MDP with discount factor α.
Relationship: neutral
Domain: NeurIPS
Sentence 1: An interesting observation about our proof technique is that the only concentration inequality we invoked was McDiarmid's Inequality (in [6,Theorem 5] to obtain (20)-a bound on the deviations in terms of the Rademacher complexity).
Sentence 2: consider f (w; θ) as in (2), where (z; θ) is convex and L-Lipschitz in z, the norm of φ(θ) is bounded by B, and r is λ-strongly convex.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Assuming that the g n are small we perform a power series expansion of ln Here we have repeatedly used the fact that each factor z n in expectations z l n z s m have to be paired (by Wick's theorem) with a factor z m where m = n (diagonal terms vanish by ( 17)).
Sentence 2: we compute explicit expressions for the remainder terms of the approximation.
Relationship: neutral