sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: 3 Nevertheless, our batch construction is designed to achieve the utmost potential of such (N +1)-tuplet loss, when using deep CNNs as embedding kernel on large scale datasets both in terms of training data and number of output classes.
Sentence 2: the proposed N -pair-mc loss is a novel framework consisting of two indispensable components: the (N +1)-tuplet loss, as the building block loss function, and the N -pair construction, as the key to enable highly scalable training.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In yet other scenarios, correspondences are sought not between projections of the same 3D point in different images, but between semantic analogs across different instances within a category, such as beaks of different birds or headlights of cars.
Sentence 2: in its most general form, the notion of visual correspondence estimation spans the range from low-level feature matching to high-level object or scene understanding.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The cost function being minimized represents the ℓ 0 (quasi)-norm of w (i.e., a count of the nonzero elements in w).
Sentence 2: unfortunately, an exhaustive search for the optimal representation requires the solution of up to M N linear systems of size N × N , a prohibitively expensive procedure for even modest values of M and N . in practical situations there is a need for approximate procedures that efficiently solve (1) with high probability.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Player i is represented by a vertex labeled i in G. We use N G (i) ~ {I, ... , n} to denote the set of neighbors of player i in G-those vertices j such that the undirected edge (i , j) appears in G. By convention, NG(i) always includes i itself.
Sentence 2: the interpretation is that each player is in a game with only his neighbors in G. if ING(i) I = k, the matrix Mi has k indices, one for each player in NG(i) , and if x E [0, Ilk, Mi(X) denotes the payoff to i when his k neighbors (which include himself) play x.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Once a replay is done, we let the agent use a new behavior policy ⇡ 0 b to collect more episodes, add them to D, and continue to update ⇥ from the augmented D. This process repeats for multiple stages, and the model parameters learned from the previous stage will be used as the initialization for the next stage.
Sentence 2: we can focus on learning at a single stage, which was formulated in Section 2 as one of solving the optimization problem (7).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The second criterion is because, intuitively, a node i's topological position within an enclosing subgraph can be described by its radius with respect to the two center nodes, namely (d(i, x), d(i, y)).
Sentence 2: we let nodes on the same orbit have the same label, so that the node labels can reflect nodes' relative positions and structural importance within subgraphs.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, we can compute the expected distance E(D j |v j ) of an image as: This represents the expected number of modifications in v from 1 to 0 (α) or from 0 to 1 (β).
Sentence 2: over all images, we can define the expected distance E(D|v) as where α i , β i ∈ [0, 1] and ∝ rank represents that the proportionality is only related to the relative ranking of the image memorability scores, and M is the total number of images.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In other words, the image parts which most describe the domain will remain unchanged, which makes it straightforward for the discriminator D T to detect the image as a fake.
Sentence 2: the only way to find an equilibrium between generator F S→T , attention map A S , and discriminator D T is for A S to focus on the objects or areas that the corresponding discriminator thinks are the most descriptive within its domain (i.e., the horses).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Denote by m f , v f , m g and v g the means and variances of Q old for f (x n ) and g(x * n ), respectively.
Sentence 2: then, EP only requires five quadratures to update each γ n . One to compute log Z n and four extras to compute its derivatives with respect to m f , v f , m g and v g . After convergence, Q can be used to approximate predictive distributions and the normalisation constant Z can be maximised to find good values for the model's hyper-parameters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We achieve this by minimizing the cross-entropy for sgm(r(x; φ)) across the training instances: The observation 2 says that z \t = 0 → z t = 1 given y.
Sentence 2: simply ignoring the case z \t = 0 and fixing q(z t |x, y; φ) = Ber(z t ; 1) is a close approximation of p(z t |x, y), especially under mean-field assumption (see the Appendix A for justification).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We can vary our λ on Liver to recover the same accuracy and solution as Shogun with a further decrease in our training time.
Sentence 2: another very positive thing is that SMO-MKL appears to be relatively stable across a large operating range of p. The code is, in most of the cases as expected, fastest when p = 2 and gets slower as one increases or decreases p. Interestingly though, the algorithm doesn't appear to be significantly slower for other values of p. it is hoped that SMO-MKL can be used to learn sparse kernel combinations as well as non-sparse ones.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For a diagonal covariance matrix, strong generalization along a particular dimension would be produced if the covariance matrix has a high variance along that dimension, but low variances along the remaining dimensions.
Sentence 2: this model can learn to strongly generalize along one dimension, but people often make strong generalizations along multiple dimensions [5], such as in Equation 2 when r < 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, we want to pick the policy that leads to the largest possible region of attraction according to Theorem 2.
Sentence 2: this requires that for each discrete state in X τ the corresponding state-action pair under the policy must be in the set D n . we optimize the policy according to The region of attraction that corresponds to the optimized policy π n according to (3) is given by V(c n ), see Fig.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To explain, we first note that optimizing the second term of R y w.r.t G actually enforces the structure information to be fully persevered in y, because C is asked to recover the structure y from G(y, z), which is generated conditioned on y, regardless of the uncertainty of z (as z is marginalized out during sampling).
Sentence 2: minimizing R y indicates the following constraint: min C,G E y∼p(y) p c (y|G(y, z 1 )), p c (y|G(y, z 2 )) , ∀z 1 , z 2 ∼ p(z), where a, b is some distance function between a and b (e.g.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The total posterior probability of an outbreak, on the other hand, will still be higher for outbreak than non-outbreak days, so the "tot" methods can perform well on OTC as well as ED data.
Sentence 2: our main result is that the Bayes tot method, which infers baselines from past counts and uses total posterior probability of an outbreak to decide when to sound the alarm, consistently outperforms the frequentist method for both ED and OTC datasets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 2 shows that the log-softmax has at most d + 1 linearly independent output vectors, even if the various inputs are applied to the model.
Sentence 2: if the vectors of true log probabilities logP * (y|x) have more than d + 1 linearly independent vectors, the softmax-based model cannot completely represent the true probabilities.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We show that Foveated Feature Congestion (FFC) clutter scores (r(44) = −0.82 ± 0.04, p < 0.0001) correlate better with target detection (hit rate) than regular Feature Congestion (r(44) = −0.19 ± 0.13, p = 0.0774) in forced fixation search; and we extend foveation to other clutter models showing stronger correlations in all cases.
Sentence 2: our model allows us to enrich clutter perception research by computing fixation specific clutter maps.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The symmetrized Sherali-Adams polytopes for an uprooted hypergraph ∇G(V + , E + ) (as given in Definition 2), is: Definition 16.
Sentence 2: for any i ∈ V + , and any integer r ≥ 2 such that max E∈E + |E| ≤ r ≤ |V + |, we define the symmetrized Sherali-Adams polytope of order r uprooted at i to be for each collection of measures over hyperedges in L i r (∇G), there exist corresponding flippinginvariant, locally consistent measures on sets of size r which contain i (and their subsets).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let r i = y πi y i and Since y and y ′′ disagree on exactly |k − k ′′ | locations (and similarly for y π and y ′′ π ), the number of locations where r and r ′′ disagree is therefore at most 2|k − k ′′ |.
Sentence 2: for any r ′′ and any h ∈ H, We observe that where z i = y i − y ′′ i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that the algorithm specifies α := λ 3 and generates a sequence of evenly-spaced indicies b t i := nα(i For every i ∈ 0..  10), ( 11) and Lemma 1, there exists some N 1 such that for all n ≥ N 1 we have, Since λ ≤ λ min , we have α ∈ (0, λ min /3].
Sentence 2: for every t = 1, 2 and every change point π k , k ∈ 1..κ we have Define µ min := min (t,i)∈I ζ(t, i).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: And with s = jω, our synthesized BM design (passive) is where the two first-order systems are both low-pass filters (LPFs), with time constants τ 1 and τ 2 , respectively; b is a gain factor.
Sentence 2: i in can be expressed in terms of I mem as: Comparing this expression with the design target (Equation 3) yields the circuit analogs: Note that the mass M (x) is a constant (i.e., 1), which was also the case in our mathematical model simulation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Recall that λ ě λ max yields βpλ,Ωq " 0, in which case ρ :" y ´X βpλ,Ωq " y is the optimal residual and y{λ max is the dual solution.
Sentence 2: as for getting λ max " Ω D pX J yq, the scaling computation in (9) requires a dual norm evaluation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Otherwise, for any i such that ↵ ⇤ i = 0 it follows that ✓ i 0, which implies  i .
Sentence 2: for any nonzero weight ↵ ⇤ i > 0 the following holds: Squaring and summing Equation ( 2) over all the nonzero entries of ↵, we arrive at the following equation for : Algorithm 1 k ⇤ -NN Input: vector of ordered distances 2 R n , noisy labels y 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The objective function of the embedding problem is then given by [17].
Sentence 2: the optimal embedding is found by solving argmin v T Dv=1 v T (D − A)v. Setting u = D 1/2 v, this is solved by finding the eigenvector corresponding to the smallest eigenvalue of L . Lemma 1 provides an efficient dimension reduction framework that summarizes the information in the entire network into few feature vectors.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, assuming the condition number 1/λ is smaller than the data size (assuming each f i (•) is O(1) smooth, and λ is the strong convexity parameter of F (•)), only O(m log(1/ )) gradient evaluations are required to get an -accurate solution, for any .
Sentence 2: we can get a high-accuracy solution after the equivalent of a small number of data passes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Theorem 1 provides a desired value h ⇤ = "/4 C, where we recall that C is the Lipschitz parameter of the optimal action-value function Q ⇤ (see Lemma 1).
Sentence 2: we need to use a small h ⇤ if we demand a small error ", or if Q ⇤ fluctuates a lot with a large C. Theorem 1, combined with Proposition 1, immediately yield the following bound that quantify the number of samples required to obtain an "-optimal action-value function with high probability, if the sample path is generated per the uniformly random policy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, the only possible consistent domain is the singleton {s a,i } or {u a,i } (i.e., P(M s i→a = 1, M u i→a = 0) + P(M s i→a = 0, M u i→a = 1) = 1).
Sentence 2: either 0 or 1 can possibly be a consistent value assignment, but not both.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By regularity condition 3a, and assuming a = 0, we have that a((∂/∂θ)f θ0 (x)) ⊤ , Z is a continuous real random variable for any x, unless Z = 0 in which case the lemma is trivial.
Sentence 2: the conditions of Lemma D.3 apply: the two sides of Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The linear embeddings are defined as follows: ( The embedding transforms W θ , W φ , and W g are all learnable and have the shape of m × l, m × l, m × m, respectively.
Sentence 2: the proposed non-local operation can be written as where δ i (X) = j∈Si φ(X i , X j ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, there are no write-after-write races (handling these is possible, but complicates the analysis).
Sentence 2: notice that, if we let x t denote the value of the vector x in the central store S after t writes have occurred, then since the writes are atomic, the value of x t+1 is solely dependent on the single thread that produces the write that is serialized next in S. If we let Gt denote the update function sample that is used by that thread for that write, and v t denote the cached value of x used by that write, then Our hardware model further constrains the value of ṽt : all the read elements of ṽt must have been written to S at some time before t. for some nonnegative variable τi,t , ) where e i is the ith standard basis vector.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, we set L V = V 2 /2, and calculate the gradient: We can simplify this equation considerably by observing that the contribution of the second summation is largely averaged out under a wide variety of realistic conditions 4 .
Sentence 2: it can be neglected, and we obtain the following local learning rule: This is a Hebbian plasticity rule, whereby connectivity changes in proportion to the presynaptic firing rate ōj and post-synaptic membrane potential V i .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: GPs with SE, MA, and RQ kernels are all truly Bayesian nonparametric models -these kernels are derived from infinite basis function expansions.
Sentence 2: as seen in Figure 1 g), h), i), these methods are completely able to capture the information in the training region; however, these kernels do not have the proper structure to reasonably extrapolate across the missing region -they simply act as smoothing filters.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is defined with respect to the following set of invertible 2 × 2 matrices G ab for {a, b} ∈ E, coined gauges: The GM, gauge transformed with respect to G = [G ab , G ba : {a, b} ∈ E], consists of factors expressed as: Here one treats independent x ab and x ba equivalently for notational convenience, and {G ab , G ba } is a conjugated pair of distinct matrices satisfying the gauge constraint G ab G ba = I, where I is the identity matrix.
Sentence 2: then, one can prove invariance of the partition function under the transformation: GT results in the gauge transformed distribution p G (x) = 1 Z a∈V f a,G (x a ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Thus, for low interaction strengths, the MF magnetization is maximized by focusing the external field on the nodes of highest out-degree in the network, independent of b and H. To study the structure of solutions to MF-IIM at high interaction strengths, we make the simplifying assumptions that J is strongly-connected and b ≥ 0 so that Theorem 6 guarantees a stable nonnegative steady state m. For large β and an additional external field h ∈ F H , m takes the form where d in i = j J ij is the in-degree of node i.
Sentence 2: in the high-β limit, we have: where i * = arg min i (d in i + b i + h i ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Examining the posterior p( , conditioned on all other parameters, we find that most settings of a and b tend to be non-informative, especially in the case of sequential learning (discussed further in Section 5).
Sentence 2: the average sparsity level of the representation is inferred by the data itself and each sample x i has its own unique sparse representation based on the posterior, which renders much more flexibility than enforcing the same sparsity level for each sample.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In contrast, for hard tanh networks, p(q ⇤ ) = erf(1/ p 2q ⇤ ).
Sentence 2: one can always move along the critical line in the ( w , b ) plane towards the point (1, 0), thereby reducing q ⇤ , increasing p(q ⇤ ), and decreasing, to an arbitrarily small value, the prefactor 1 p(q ⇤ ) p(q ⇤ ) controlling the linear growth of both max and 2 JJ T . So unlike either ReLU networks, or Gaussian networks, one can achieve dynamical isometry up to depth L by choosing q ⇤ small enough so that p(q ⇤ ) ⇡ 1 1 L . In essence, this strategy increases the fraction of neurons operating in the linear regime, enabling orthogonal hard-tanh nets to mimic the successful dynamical isometry achieved by orthogonal linear nets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In addition to two-sample homogeneity test, inlier-based outlier detection, and transfer learning, density-ratios can be useful for tackling various machine learning problems, for example, multi-task learning, independence test, feature selection, causal inference, independent component analysis, dimensionality reduction, unpaired data matching, clustering, conditional density estimation, and probabilistic classification.
Sentence 2: it would be promising to explore more applications of the proposed relative density-ratio approximator beyond two-sample homogeneity test, inlier-based outlier detection, and transfer learning.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let C be the set of consensus decisions: then one can reformulate (3) in a decentralized way as follows: where Next, we consider the implementation of PDA in (5) to solve (11).
Sentence 2: let Φ(x) given the initial iterates x 0 , θ 0 , λ 0 and parameters γ > 0, τ i , κ i > 0 for i ∈ N , choosing D x and D y as defined in Definition 1, and setting ν x = ν y = 1, PDA iterations given in (5) take the following form: Starting from µ 0 = λ 0 , compute for i ∈ N Using extended Moreau decomposition for proximal operators, λ k+1 can be written as Let Let B {x : , where p(x) Although x-step and θ-step of the PDA implementation in ( 12) can be computed locally at each node, computing λ k+1 requires communication among the nodes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Proof: If g is any function in H such that g(x j ) = 0, j ∈ IN m and t a real number such that |t| g ≤ , for > 0, then Consequently, we have that f 0 2 ≤ f 0 + tg 2 from which it follows that (f 0 , g) = 0.
Sentence 2: f 0 satisfies and the result follows from Lemma 3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For convenience, we also include the auxiliary segment s The relation kernel computes the number of common patterns between two sentences s and t, where the set of patterns is restricted to the three types introduced in Section 3.
Sentence 2: the kernel rK(s, t) is expressed as the sum of three sub-kernels: f bK(s, t) counting the rK(s, t) = f bK(s, t) + bK(s, t) + baK(s, t) bKi(s, t) = Ki(s b , t b , 1) number of common fore-between patterns, bK(s, t) for between patterns, and baK(s, t) for between-after patterns, as in Figure 3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: While our theorem specified an optimal setting of the regularization parameter n and ⌫, this optimal setting depended on unknown model parameters.
Sentence 2: as is standard with high-dimensional regularized estimators, we set tuning parameters n = c p log p/n and ⌫ = c 0 p log p/n by a holdoutvalidated fashion; finding a parameter that minimizes the `2 error on an independent validation set.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this simple setting, we can tune the step size optimally as η * = O( (1 + P T )/T ) and obtain an improved O( T (1 + P T )) dynamic regret bound, which matches the lower bound in Theorem 2.
Sentence 2: when bounding the general dynamic regret, we face the following challenge: On one hand, we want the regret bound to hold for any sequence of comparators, but on the other hand, to get a tighter bound, we need to tune the step size for a specific path-length.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, we can invoke [5, Theorem 2 and Section 4.3] which guarantees that for some positive absolute constants C 1 , C 2 , and C 3 if (8) holds then holds for all valid B , thereby for all valid X , with probability exceeding 1 − e −C3n .
Sentence 2: with C = C 2 , the target matrix X would be feasible in (7).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If i and j are two pixels in different objects, we know they will be independent, and as such uncorrelated, resulting in zero entries in the covariance matrix.
Sentence 2: if we arrange the pixels by their object assignment, the eigenvectors of such a covariance matrix would be of the form: where v is an eigenvector of the stationary (within-object) covariance and the rest of the entries are zeros, thus eigenvectors of the covariance will be zero on one side of the occlusion mask and Fourierlike on the other side.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: EM objective) arg max w y P (x, y|w) = arg max w max qy y q y log P (x, y|w) +H(q y ) where q y is a distribution over possible missing values [10].
Sentence 2: much of the analysis we provide for joint EM also applies to marginal EM, leaving only a separate convex relaxation of the entropy term that can be conducted independently.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 2 It is often taken for granted in the optical flow literature, because in the limit where two images are sampled infinitesimally close in time, there are no occluded regions, and one can focus solely on motion discontinuities.
Sentence 2: most variational motion estimation approaches provide an estimate of a dense flow field at each location on the image domain, including occluded regions.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In instances where theoretical prediction error is available, is finite, and d = 500, the relative error between empirical and theoretical prediction error for ŷpcr and ŷbc ranges from 8.60%-33.81%; for d = 5000, it ranges from 1.72%-4.86%.
Sentence 2: the accuracy of the theoretical prediction error formulas tends to improve as d increases, as one would expect.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Motivated by this fact, the first intuition of our algorithm is that in order to minimize disaggregation error, we can discriminatively optimize the bases B 1:k that such performing the optimization (2) produces activations that are as close to A ⋆ 1:k as possible.
Sentence 2: of course, changing the bases B 1:k to optimize this criterion would also change the resulting optimal coefficients A ⋆ 1:k . the second intuition of our method is that the bases used in the optimization (2) need not be the same as the bases used to reconstruct the signals.
Relationship: reasoning