sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: While one can expect that mixtures of splines (Bar-Joseph et al., 2002) or functions with universal approximation capabilities can be fitted to any mean behavior, the restricted class of functions that we proposed (based on the simplified dynamics of the mRNA changes implied by the differential equation in Equation ( 4)) is likely to fail if the true dynamics does not match the assumptions.
Sentence 2: there are two main reasons to use the proposed restricted class of functional forms: (1) to be able to interpret the resulting mean curves in terms of the synthesis / decay rates at each of the regimes as well as the switching times; (2) to naturally incorporate alignment by real-values shifts along the time axis.
Relationship: neutral
Domain: NeurIPS
Sentence 1: The Hadamard-Rademacher process in two dimensions: explores a state-space of 16 orthogonal matrices, is ergodic with respect to the uniform distribution on this set, has period 2, the diameter of the Cayley graph of its state space is 3, and the chain is fully mixed after 3 time steps.
Sentence 2: aW acknowledges support by the Alan Turing Institute under the EPSRC grant EP/N510129/1, and by the Leverhulme Trust via the CFI.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In parallel, recent work in neurophysiology shows that the basal ganglia can be modeled by an actor-critic version of temporal difference (TD) learning [4][5] [6], a well-known reinforcement learning algorithm.
Sentence 2: the signal generated by dopaminergic (DA) neurons resembles the effective reinforcement signal of temporal difference (TD) learning algorithms [5][8].
Relationship: neutral
Domain: NeurIPS
Sentence 1: [2] showed that SVM-like algorithms optimizing the AUC have good generalization guarantees, and [11] showed that maximizing the margin of the pairs, defined by the quantity g(x) − g(x ), leads to the minimization of the generalization error.
Sentence 2: to derive the bounds, we consider φ, the 1-Lipschitz function defined by φ Given a training example z, we denote by z l its label and z f its feature representation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: When the statistical model M is regular, or the true distribution Po (x) is at a regular point , the mle-based p(x, 0) and the Bayes predictive distribution are asymptotically equivalent, and are Fisher efficient under reasonable regularity conditions, where d is the dimension number of parameter vector (j.
Sentence 2: by substituting t(w) in ( 14) the log likelihood function becomes t=l the mle w is given by the maximizer of L(w), w = argmaxwy2(w).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We define point-wise cost function c : , which measures the cost on a position k by comparing G S (x x x, k) and the gold-standard tag y y y (k) , and we introduce the point-wise loss as Then, we define sample-wise cost function C : Y n × Y n → R + , which is the cost function with respect to a whole sample, and we introduce the sample-wise loss as Given G and a training set S, what we are most interested in is the generalization risk in structured prediction (i.e., expected average loss) [18,7]: Since the distribution D is unknown, we have to estimate R(G S ) by using the empirical risk: To state our theoretical results, we must describe several quantities and assumptions following prior work [2,12].
Sentence 2: part-of-Speech Tagging (POS-Tagging).
Relationship: neutral
Domain: NeurIPS
Sentence 1: There also has been an early attempt on transfer in planning problems, for two specific classical (deterministic) domains of TSP and Sokoban [Groshev et al., 2018].
Sentence 2: a 'good' state embedding space is a space in which similar states are close together and dissimilar states are far apart (similarly for state-action pair embeddings).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Experimental Settings: Posterior inference for the truncated models (i.e., EPM, CEPM, and DEPM) were performed using standard (non-collapsed) Gibbs sampler.
Sentence 2: as the Gibbs sampler for φ and ψ can be derived straightforwardly, the posterior inference for all parameters and hyperparameters of the DEPM also can be performed via closedform Gibbs sampler (detailed in Appendix C).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The 18 conditions come from using c = ±1 and c = ±0.3, with the degraded condition (|c| = 0.3) only being run for the case of low load, as in figure 1D.
Sentence 2: to take account of the possibility of erroneous, early, responses, there is also a probability of 0.01 per step of stopping the accumulation and reporting whichever sign of target has a higher probability (guessing randomly if this probability is 0.5).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The contingency awareness approach [4] improved performance of the SARSA algorithm by augmenting the feature sets with a learned representation of the parts of the screen that are under the agent's control.
Sentence 2: finally, recall that the UCTtoClassification-Interleaved agent was designed so that its input distribution during training is more likely to match its input distribution during evaluation and we hypothesized that this would improve performance relative to UCTtoClassification.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We next examined how well the proposals explore the region assignment of clusters.
Sentence 2: for any measure µ over Θ ⊗ Y define the restricted projection µ t by Note that µ t is a measure over Θ for each t ∈ T. Now let α be a base measure over the product space Θ ⊗ Y and consider a gamma process over Θ ⊗ Y.
Relationship: neutral
Domain: NeurIPS
Sentence 1: By the assignment rule, we have A t xt ∈ ∂σ t (ŷ t ), and the dual variable update can be viewed as ŷt+1 ∈ argmin y t s=1 A s xs , y − ψ * (y).
Sentence 2: the algorithm is stated and analyzed for a twice differentiable, separable ψ(u).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Hence, in our analysis we need to separate out the eigenvectors that correspond to well spaced eigenvalues from the others.
Sentence 2: . . , u d be the eigenvectors of G. We define the j-th eigenvalue separation ∆ j by For any ∆ > 0 define also Our results are expressed in terms of the effective rank ( 6) of G at horizon T . in order to account for the error in estimating the eigenvalues of G, we define the effective rank ρ t with respect to the following slight variant of the function kappa, Let M (i) be the estimated gradient outer product constructed at the end of phase i, and let µ and u 1 (i), .
Relationship: neutral
Domain: NeurIPS
Sentence 1: In such downstream models, latent topic assignments are sufficient statistics for the prediction model and it is easy to define the max-margin constraints based on existing max-margin methods (e.g., SVM).
Sentence 2: by using a normal prior on ψ, the scene model can capture the mutual correlations between different objects, similar to the correlated topic models (CTMs) [3].
Relationship: neutral
Domain: NeurIPS
Sentence 1: We will refer to this as the naive group lasso approach.
Sentence 2: . . , X k n k ∈ R p are independent and identically distributed from a N (0, Σ k ) distribution, for k = 1, . . . , K. Letting S k denote the empirical covariance matrix for the kth class, one can maximize the penalized log likelihood maximize where , λ 1 and λ 2 are nonnegative tuning parameters, and ) is a penalty applied to each off-diagonal element of Θ 1 , .
Relationship: neutral
Domain: NeurIPS
Sentence 1: The regularized permutahedron-based convex formulation is a convex QP with O(m) variables and constraints, where m is the number of comparators in its sorting network, while the Birkhoff-based one is a convex QP with O(n 2 ) variables.
Sentence 2: in this paper, we use a relaxation based on these polytopes, which we call "sorting network polytopes."
Relationship: neutral
Domain: NeurIPS
Sentence 1: The squared Euclidean distance between two instances x i and x i is ∑ j∈N (x ij − x i j ) 2 , while the weighted squared Euclidean distance is written as where w := [w 1 . . . w ] ∈ [0, 1] is an -dimensional vector of non-negative weights and ε i,i := We develop a feature weighting algorithm within the framework of regularized empirical risk minimization, i.e., minimizing the linear combination of a loss term and a regularization term.
Sentence 2: the number of neighbors k ∈ {1, 3, 5} was selected based on the classification performance on the validation set.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Let y = (Y1, Y2 , ... ,Yn)T be the map from the graph to the real line.
Sentence 2: it fo llows from the Stokes theorem that -div and V are form ally adjoint operators, i. e. if f is a function and X is a vector field fM (X, V f) = fM div(X)f. Thus We see that .c is positive semidefinite and the f that minimizes fM IIV fl12 has to be an eigenfunction of .c.
Relationship: neutral
Domain: NeurIPS
Sentence 1: This analytical finding suggests the following generic, iterative procedure to obtain characterizers that minimize model variance: 1.
Sentence 2: given an OOM A = (R m , (τ a ) a∈O , w 0 ) with an induced probability distribution P A , its reverse OOM A r = (R m , (τ r a ) a∈O , w r 0 ) is characterized by a probability distribution P A r satisfying A reverse OOM can be easily computed from the "forward" OOM as follows.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Then we have which is a standard result from the singular value decomposition (SVD) [12].
Sentence 2: as some concrete examples, we can construct a large family of clipped µ 1 L q + µ 2 l 2 2 where 0 < q ≤ 1 by restricting f ′ i = w i I(|w i | < ǫ) + c for some constant ǫ and c. Also, SCAD [9] satisfies all three conditions so it belongs to f * .
Relationship: neutral
Domain: NeurIPS
Sentence 1: This line of research is orthogonal to our work since their analyses are regarding to the convergence of the distribution of the iterates to the stationary distribution of Langevin diffusion in total variation distance or 2-Wasserstein distance instead of expected function value gap.
Sentence 2: bounding I 2 : Convergence to Stationary Distribution of Langevin Diffusion Now we are going to bound the distance between two invariant measures µ and ⇡ in terms of their expectations over the objective function F n . Our proof is inspired by [51,12].
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our goal is to cluster these multi-dimensional time-series automatically into a small number of motion templates without pre-labeling of the trajectories or assuming an a priori number of templates.
Sentence 2: we will first introduce a generative approach for clustering and modeling multi-dimensional time-series with Bayesian Mixtures of LGSSMs and describe how this approach can be made tractable using a variational approximation.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Therefore, Finally, for any b ∈ ∆(∼ Γn ) and s ∈ S with b(s) = 1, and any other state ŝ with b(ŝ) = 1, it must be the case that s ∼ Γn ŝ and T n (0)(s, ŝ) = 0.
Sentence 2: we can use this result in the definition of local refinement Γ j−1 ζj Γ j to conclude that s ∼ Γj−1,ζj s . Moreover, as ŝ This completes the definition of consistency on η, and it becomes clear that s ∼ Γn−1,ζn s ⇒ s ∼ Γ0,η s , or ∆(∼ Γ0,η ) ⊆ span(∆(∼ Γn−1,ζn )).
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the legend, the estimated slopes of ∆R π (T ) (in logarithmic scale) are indicated after the policy names.
Sentence 2: assuming, for example that µ 1 = µ 2 + ∆, then an ETC strategy π chooses the suboptimal arm We denote the set of all ETC strategies by Π ETC .
Relationship: neutral
Domain: NeurIPS
Sentence 1: This makes computing even the expected value of max α β A,α an intractable task, let alone obtaining concentration of measure results that could give us upper bounds on the probability of condition 14 holding under the assumed distribution on A.
Sentence 2: mLKNN's two parameters k (number of neighbors) and s (strength of the uniform prior) were kept fixed to 10 and 1.0, respectively, as was done in [20].
Relationship: neutral
Domain: NeurIPS
Sentence 1: The sociability and receptivity factors, referred to as control parameters, support a clear picture of the asynchronous, synchronous aperiodic and periodic synchronous patterns.
Sentence 2: an asynchronous aperiodic regime (region B) is prone to evolve into an asynchronous (region A) or periodic synchronous (region C) regimes (Figure 6).
Relationship: neutral
Domain: NeurIPS
Sentence 1: The asymptotic expansions in [43,37,23,40] likewise do not provide a computable finite-sample bound.
Sentence 2: that algorithm suffers from the cost of cross-validating over both the dimension d and number of neighbors k. Also, the technique is only applicable in Hilbert spaces (as opposed to more general metric spaces) and provides only asymptotic consistency, without finite-sample bounds such as those provided by KSU.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Sequential effects have been demonstrated in domains as varied as legal reasoning and jury evidence interpretation (Furnham, 1986;Hogarth & Einhorn, 1992) and clinical assessments (Mumma & Wilson, 2006).
Sentence 2: because responses are influenced by recent context, they are not as pure a measure of an individual's stimulus appraisal as one might wish for.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We have evaluated seven nowcasting algorithms, including the simplest model which always predicts the last frame, two optical flow based methods (ROVER and its nonlinear variant), and four deep learning methods (TrajGRU, ConvGRU, 2D CNN, and 3D CNN).
Sentence 2: in this paper, we have provided the first large-scale benchmark for precipitation nowcasting and have proposed a new TrajGRU model with the ability of learning the recurrent connection structure.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Furthermore, complex Ising models have found widespread use in machine learning, and our model is formally equivalent to a pair-wise Markov random field or a Boltzmann machine [18][19][20].
Sentence 2: in the next section we show that if there exists a stable and entry-wise non-negative steady-state, and if J is strongly-connected, then M M F (b + h) is both smooth and concave in h, regardless of the interaction strength.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Proof of of Corollary 2.2: We first observe that we have Obviously, the inclusion "⊂" directly follows from the above monotonicity.
Sentence 2: given a function f : X → R, Lemma 2.3 shows that for P X -almost all x ∈ X there exists a unique t * (x) is the element in M * (x) that has the smallest distance to f (x).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since γ * 2 can be expressed as the optimum of semidefinite program, we can find in polynomial time a probability distribution P and a weight function Q on S such that γ * 2 (P − Q) is minimizd.
Sentence 2: the analysis of random samples does not help us here, because these proofs do not reveal the structure that makes generalization possible.
Relationship: neutral
Domain: NeurIPS
Sentence 1: A sample of the results is shown in Figure 4; as can be seen, the proposed algorithms UCB-TC, UCB-UC, and UCB-BA generally outperform existing baselines in terms of minimizing regret relative to the top cycle, the uncovered set, and the Banks set, respectively.
Sentence 2: for the three tournament solutions discussed above, we will define the following natural notions of individual regret: ∈ TC(P) In the special case when P admits a Condorcet winner i * , the three individual regrets above all reduce to the Condorcet individual regret, r CW P (i) = ∆ P i * ,i .
Relationship: neutral
Domain: NeurIPS
Sentence 1: In addition, we show that in the stochastic setting SODA achieves an O a:∆a>0 Kε 2 ∆a pseudo-regret bound that holds simultaneously with the adversarial regret guarantee.
Sentence 2: first we provide an upper bound for the expected regret of SODA against oblivious adversaries that produce loss sequences with effective loss range bounded by ε.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Since the αs are non-negative, and since there are ties neither the αs nor in the βs, the rearrangement inequality implies that the minimum value of R (p σ , σ ) is obtained for the single permutation σ for which the β σ -1 (k) are in reverse order relatively to the α σ -1 (k) (i.e.
Sentence 2: [11] showed that many convex losses based on pairwise comparisons, such as those of RankBoost [12] or Ranking SVMs [14,4], are not calibrated with respect to the PD.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Provided the generative model is stationary (i.e., x 0 = 0 and Q 0 = Π), SSID algorithms yield consistent estimates of the parameters A, C, Q, R, d without iteration.
Sentence 2: an alternative is the spectral approach known as subspace identification (SSID) in the engineering literature [3,4,5].
Relationship: neutral
Domain: NeurIPS
Sentence 1: In the design of this algorithm and its performance analysis, we face two challenges: (i) we only have access to a sampled version A (B) of M (B) ; and (ii) U Σ(V (B) ) is not the SVD of M (B) since the column vectors of V (B) 1:k are not orthonormal in general (we keep the components of these vectors corresponding to the set of indexes B).
Sentence 2: to translate our bound (1) in an additive error is easy: Sparsifying M to speed-up the computation of a low-rank approximation has been proposed in the literature and the best additive error bounds have been obtained in [AM07].
Relationship: neutral
Domain: NeurIPS
Sentence 1: [17], who show that that locally private algorithms coincide with concepts that can be learned with polynomial sample complexity in Kearns's statistical query (SQ) model.
Sentence 2: indeed, if we can find P * and Q * satisfying the saddle point (4), then the trivial direction of the max-min inequality yields To fully formalize this idea and our notions of privacy, we define two collections of probability measures and associated losses.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In CS-MRI, it commonly needs to run the ADMM algorithm in dozens of iterations to get a satisfactory reconstruction result.
Sentence 2: 2 Deep ADMM-Net for Fast MRI General CS-MRI Model: Assume x ∈ C N is an MRI image to be reconstructed, y ∈ C N ′ (N ′ < N ) is the under-sampled k-space data, according to the CS theory, the reconstructed image can be estimated by solving the following optimization problem: where is a under-sampling matrix, and F is a Fourier transform.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 3, we empirically confirm that the recall value of geometrically related predicates are significantly increased, such as using, carrying, riding.
Sentence 2: compared to previous models [33,18] that have been proposed to focus on message passing between nodes and edges, our model explicitly reasons about the relations within nodes and edges and predicts graph elements in multiple steps [35], such that features of a previous stage provides rich context to the next stage.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In this paper we have introduced Sobolev Training for neural networks -a simple and effective way of incorporating knowledge about derivatives of a target function into the training of a neural network function approximator.
Sentence 2: [30] and antecedents propose a technique for Bayesian optimisation with Gaussian Processess (GP), where it was demonstrated that the use of information about gradients and Hessians can improve the predictive power of GPs.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that c/2 is the expected degree of the G ER (p, c/p) ensemble.
Sentence 2: similarly, for both the Erdős-Rényi graph ensemble G ER (p, c/p) and small-world ensemble G Watts (p, d, c/p), we can establish that the threshold J * = Θ(1/c), and thus, the observations made for the Gaussian setting hold for the Ising model as well.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Throughout the paper, the measures P o and P 1 assign probability measures to the paths X τ (t)|Θ = 0 and X τ (t)|Θ = 1 respectively, and we assume that P o << P 1 3 .
Sentence 2: we conclude our analysis for the process µ t by noting that lack of information samples creates bias towards the belief that Θ = 0 (e.g.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Our approach satisfies the error bound tighter and reduces computational cost because: • Each cluster's maximum truncation number no longer depends only on its farthest point, so if most points are clustered close to the center the maximum truncation will be lower; • The weight of each source point is considered in the error contributions, so if a source point is far away but has a weight of q i = 0 its error contribution will be ignored; and finally • Each target can use a truncation number that depends on its distance from the cluster.
Sentence 2: for each dataset we ran five experiments: the first four fixed one of the four methods (direct, direct+tree, ifgt, ifgt+tree) and used it for all conjugate gradient iterations; the fifth automatically selected the best method at each iteration (denoted by auto in figure 6).
Relationship: neutral
Domain: NeurIPS
Sentence 1: Note that h(⌧ ) is continuous and strictly decreasing, with h(0) = ⇡ 1 and h(1) = ⇡ 1 .
Sentence 2: interestingly, OFO seems to properly adjust the values 1/b t+1 in an adaptive manner (b t is a sum of two terms, the first of which is t⇡ 1 in expectation), which is a very nice property of the algorithm.
Relationship: neutral
Domain: NeurIPS
Sentence 1: Here, different subspaces may correspond to motion of different objects in video sequence [11,17,20], different rotations, translations and thickness in handwritten digit or the latent communities for the social graph [15,5].
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: neutral
Domain: NeurIPS
Sentence 1: The ventral striatum involved in reward prediction error δ(t) on shortest time scale (γ = 0), while the dorsolateral striatum correlated with reward prediction V(t) on longer time scales (0.9 ≤ γ ≤ 0.99).
Sentence 2: in order to maximize the reward acquired in a long run, the subject has to select an action by taking into account both the immediate reward and the future reward expected from the subsequent state.
Relationship: neutral
Domain: NeurIPS
Sentence 1: 1 defines the likelihood maximization problem, which is iteratively solved using EM.
Sentence 2: an approach for exploiting embedded structure in the transition and observation functions is to replace the exact forward-backward message computations with exact forward filtering and backward sampling (FFBS) [11] to obtain a sampled reverse trajectory consisting of �s T , n T i,l , a T i �, �n T −1 i,l , a T −1 i , o T i , n T i,l �, and so on from T to 0.
Relationship: neutral
Domain: NeurIPS
Sentence 1: We used mini-batches of size 100 and the step-size of 10 −α , where α is an integer between 0 and 10.
Sentence 2: to see this, we consider the vector π(w), the path vector, where the number of coordinates of π(w) is equal to the total number of paths from the input to output units and each coordinate of π(w) is the equal to the product of weights along a path from an input nodes to an output node.
Relationship: neutral
Domain: NeurIPS
Sentence 1: In recent years, there has been a trend towards unified statistical analyses that provide statistical guarantees for broad classes of models via general theorems [12].
Sentence 2: given the joint distribution in (10), the conditional distribution of X s given the rest of the nodes is given by: Let θ * \s = {θ * st } t∈V \s ∈ R p−1 be a zero-padded vector, with entries θ * st for t ∈ N (s) and θ * st = 0, for t ∈ N (s).
Relationship: neutral
Domain: NeurIPS
Sentence 1: We would like to differentiate W U with respect to U .
Sentence 2: we now derive formulas for the first and second order derivatives of f . In deriving these formulas, it is useful to note that, for a suitably smooth mapping g, grad X → 1/2 g(X) 2 where Dg(X) * is the adjoint of the differential of g at X.
Relationship: neutral