sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Bounding the expected variance by the empirical variance is generally tighter than bounding it by the empirical loss.
Sentence 2: the PAC-Bayes-Empirical-Bernstein bound is generally tighter than the PAC-Bayes-kl bound, although the exact comparison also depends on the divergence between the posterior and the prior and the sample size.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The proof of Theorem 3 relies on two facts.
Sentence 2: first, under H 0 , Σq = Σp i.e., the plug-in estimate of Σ p . under H 0 , the null distribution approximated with Σq is asymptotically correct, following the convergence of Σp to Σ p . Second, the rejection threshold obtained from the approximated null distribution is asymptotically constant.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , σ ′ p where, for all j = 1, . . . , p, σ ′ j has the same support as σ j .
Sentence 2: we note that unlike in the symmetric case, the signs of the off diagonal entries can no longer be determined using a cycle basis of induced cycles, since such a basis may contain only cycles which have no contribution to the principal minors of K. In the same example as above, the only induced cycles of G K are triangles, and any cycle basis should contain at least three cycles.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 9) and write, The squared norm, • 2 is a convex function in its vector argument and thus ∆ t is concave, which yields the following lower bound on ∆ t , The SimPerc algorithm sets µ t j to be 1/|M t | for all j ∈ M t and to be 0 otherwise.
Sentence 2: furthermore, for all j ∈ M t , α t j is set to C. the right hand-side of Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Figure 1(a) shows the initial discretization of our example domain into 12 large cells together with the start coordinates of the agent (A) and the goal region (cell containing ¢¡ ¤£ ¦¥ ¨ § ).
Sentence 2: it can always attempt to move towards the center of each adjacent cell (that is, cell that its current cell shares a border line with).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then the rest of id T = 4 nodes are changed to have id T = 3 by Line 15 of Algorithm 1 and to be further processed (even though they could be both parents at the same time) with neighbor nodes' causal identities.
Sentence 2: step 2 of Algorithm 1 makes all variable pairs with id T = 4 to become identified either as parents, children, or with id T = 3 after taking some neighbors' MBs into consideration.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Or, by iteratively proposing a ξ t taking into account the last state h t−1 proposed and the generative deterministic mechanism F g . The latter allows the inference network to peek at states h t that F g could generate from h t−1 before proposing an actual target h t . This allows the inference model to track a multi-modal F g without need for F q to match its expressiveness.
Sentence 2: this might offer the possibility to learn multi-modal generative models, without the need to employ complex multi-modal distributions in the inference model.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Hence, an intervention that blocked the WNT5A protein from activating its receptor could substantially reduce the ability of WNT5A to induce a metastatic phenotype.
Sentence 2: we consider the following immediate reward function in belief space: R(s, = 50 128 i=1 s(i) δ x i (1)=0 − 10||a|| 1 . actions are available for controlling the system: A = {[0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 0, 1, 0]}.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: . . , N K and drawing µ from an infinite-dimensional Dirichlet distribution over the positive integers: where α > 0 is a fixed concentration parameter and µ (0) = (µ m ≥ 0 for all m. The probability of C N conditioned on N and µ is Posterior inference for the NBD model involves alternating between (a) sampling C N from P (C N | N, a, q, µ) using the chaperones algorithm and (b) sampling µ from its conditional posterior: where L m is the number of clusters of size m in C N . Although µ is an infinite-dimensional vector, only the first N elements affect P (C N | a, q, µ).
Sentence 2: it is sufficient to sample the (N + 1)-dimensional vector (µ 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In the extreme case when µ = ∞, the second term in ( 7) can be shown to be the generalization bound using the weighted samples in the source domain only (the target data will not be effective in this case), and the last term equals to 2λ α,β + d H∆H (D α,β , D T ).
Sentence 2: effective transfer is possible in this case only if the divergence is small.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The above equation then requires which implies that where .
Sentence 2: taking union bounds on all matrix entries, we have P (7) does not hold < (p + 5p 2 )e −Cn + 2p 2 e −Cn/ν < (7 + 1 n )p 2 e −Cn/ν 2 , where the second inequality is due to the fact that p > n and ν > 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, as discussed earlier, the analysis of [14] shows that when SNR • MAR → ∞, lasso requires m > 2k log(n − k) + k + 1 (14) for reliable recovery.
Sentence 2: like thresholding, lasso does not achieve a scaling better than m = O(k log(n − k)), even at infinite SNR.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In other words, all Z ∈ Ω 1 , forms the set of extreme points of Γ 1 .
Sentence 2: proof: It is easy to check that any convex combination i θ i Z i , such that Z i ∈ Ω 1 , resides in Γ 1 , with the following two facts: (1) a convex combination of PSD matrices is still a PSD matrix; (2) all eigenvalues of Z must satisfy: .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The rest is obtained by backpropagation.
Sentence 2: the derivative of the loss with respect to the network's output is: expanding the right hand side and denoting C = 1/ √ 2π we have: Equation 9 provides a simple analytical formula for computing the gradient of Houdini with respect to its input, hence an efficient way to obtain the gradient with respect to the input of the network x by backpropagation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Furthermore, since X ∞ (αI) converges to X, the scalars , where λ k (•) is defined as the eigenvalue corresponding to eigenvector v k and not necessarily the k th largest eigenvalue.
Sentence 2: for all k such that λ k ( X) > 0, by the continuity of log, we have Defining ν(β) = s∞(β) /β, we conclude that for all k such that λ Thus, for every ∈ (0, 1], for sufficiently large β we have shown that lim β→∞ A * (ν(β)) I and lim β→∞ A * (ν(β)) X = X establishing the optimality of X for (5).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: At iteration t of the update rule we make a call to the black box ✓(x t ) giving a -approximation to the backward mapping ✓(x t ), compute x t s ✓(x t ), and then project onto [q✏, 1) p . Proposition 4.6 ensures that x t+1 2 M 1 .
Sentence 2: the update x t+1 = P (x t s ✓(x t )) is the same as x t+1 = P M1 (x t s ✓(x t )).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To summarize, from the small enclosing subgraphs extracted around links, we are able to accurately calculate first and second-order heuristics, and approximate a wide range of high-order heuristics with small errors.
Sentence 2: given adequate feature learning ability of the model used, learning from such enclosing subgraphs is expected to achieve performance at least as good as a wide range of heuristics.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Support vector machines are arguably amongst the most successful classification algorithms of all times, however their multi-class extensions such as one-vs-all [4] or clever refinements of the loss functions [10,21] all require at least one weight vector per class label.
Sentence 2: the MTL adaptation of svm [15] requires all tasks to share an identical set of labels (or require side-information about task dependencies) for meaningful tranfer of knowledge.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Intuitively speaking, such q init already encodes certain preferences towards a few preconditioned shift truncations of the ground truth.
Sentence 2: we randomly choose an index i and set the initialization point as For a generic kernel a 0 ∈ S k−1 , A T A is close to a diagonal matrix, as the magnitudes of off-diagonal entries are bounded by column incoherence µ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: These differences yield an important consequence: VDMGP can infer automatically the latent dimensionality K of data, but SPGP-DR is unable to, since increasing K is never going to decrease its likelihood.
Sentence 2: vDMGP follows Occam's razor on the number of latent dimensions K. We will assess VDMGP on real-world datasets.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Now, it is easy to check that d ρ (x, x ) := ρ(x) − ρ(x ) H defines a metric that generates ρ −1 (τ V ) and, since ρ is isometric with respect to this new metric, we can conclude that (X, d ρ ) is a compact metric space.
Sentence 2: we are back in the situation of Theorem 2.2, and hence k X is universal with respect to the space C(X, d ρ ) of functions X → R that are continuous with respect to d ρ .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, as the size of the test candidates n gets bigger, which is happening nowadays, the number of required pairwise labeling grows exponentially O(n 2 ) leading to the unfeasible implementation.
Sentence 2: there is an urgent need to reduce the number of pairwise comparisons, that is, selecting part of the pairs but without loosing the aggregation accuracy.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From Lemma 2, the near optimal auxiliary function ρ * can be regarded as minimizing the mean squared residuals to zero between R(s, a) and the estimator R ρ (s, a; ω) ∇ θ ln{d θ (s) ι π(a|s;θ)} ω + ρ(s, s +1 ).
Sentence 2: the meaning of this near optimality of g * ι,θ (s, a; ω) is interpreted as a near minimization of the Euclidean distance between r and its approximator R ρ * (s, a; ω), so that ρ * works to reduce the distance of the regressand r and the subspace of the regressor ∇ θ ln{d θ (s) ι π(a|s;θ)} of the M-estimator ω.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The dynamics of such collective computations can be seen in the coordinated activity of all of the neurons within the local network; although each individual such neuron may reflect this coordinated component only noisily.
Sentence 2: we hope to identify the computationallyrelevant network dynamics by purely statistical, unsupervised means-capturing the shared evolu-tion through latent-variable state-space models [1,2,3,4,5,6,7,8].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, a discrete time drift diffusion process can be derived by subtracting the likelihoods of Eq.
Sentence 2: ( 5): the change in y within a trial, in which the true stimulus is constant, is Gaussian: δy ∼ N (2r/δt, 4r 2 σ 2 /δt).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To analyze the computational complexity of each SGD step, we note that Hamming distance between g (defined in (10)) and h = sgn(W x) is bounded above by the depth of the tree d. This is because only those elements of g corresponding to the path to a selected leaf can differ from sgn(W x).
Sentence 2: for SGD the expression ( g − h) x T needed for Line 6 of Alg 1 can be computed in O(dp), if we know which bits of h and g differ.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From a Bayesian perspective, optimization of either problem can be viewed, after a exp[−(•)] transformation, as a challenging MAP estimation task with a quadratic likelihood function and a prior that is both improper and discontinuous.
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 methods that efficiently solve (1) or (2) with high probability.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The where and what modules are connected by two links: first, the input to the unsupervised path of the what module is the generated layout from the where module; second, we apply the same affine transformation parameters to the generated shape, so that it can be inserted to the same location as being predicted in the where module.
Sentence 2: the final output is obtained as follows: where Â and s are generated from the generator of the where and the what modules, respectively: The STN is able to make a global connection between x from the input end, and x in the output end.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: When many such maximizing sets exist, we assume that the latent options are one particular set from amongst these choices; for discussion, the particular choice does not matter, but it is important to have a single set.
Sentence 2: our goal is to recover the termination sets of the latent options from the sampled subgoal data; these can be used to construct a library of options that approximate the latent options and have the following desirable properties: • Recall: The termination sets of the library options should contain a maximal portion of the termination sets of the latent options.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: With respect to the learning rule, the pseudo-inverse is close to SVM rule in the regime we work, so we believe that is a good tradeoff between realism and tractability.
Sentence 2: although the unavoidable simplicity of our model, we believe its analysis yields important insights which will likely carry over to the more realistic domains of deep networks studied in ML and neuroscience.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: It is not hard to see that problem ( 16) is a special case of the general rank minimization problem (2) with f (X) ≡ 0, ν = 1, Ω = m×n , and X = {X ∈ m×n : ij = M ij , (i, j) ∈ Θ}.
Sentence 2: the PD method proposed in Subsection 2.2 for problem (2) can be suitably applied to (16).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, performing elementwise operations takes nearly the same amount of time on the GPU whether operating on tens or on thousands of elements, and multiplying a few hundred different vectors by the same matrix is significantly slower than executing a single (equivalent) matrix-matrix product using an optimized GEMM implementation on either a GPU or a CPU.
Sentence 2: careful grouping of operations into batches that can execute efficiently in parallel is crucial for making the most of available hardware resources.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In order to obtain a low spectral radius of 1m +o:K this interval should lie symmetrically around zero, which is equivalent to ),min = -),max' This yields 0:* = 1 >'=in l ~I >'=ax l < k with H = IAmaxl.
Sentence 2: 0:* leads to convergence according to Theorem 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In particular, if G µ (B) is the optimal value of some perturbed density restricted to B, then the event that the optima over ⌦ is contained in B is equivalent to the event that G )) is a probability measure proportional to µ(B) [12].
Sentence 2: we can use the Gumbel process for a continuous measure µ(B) = R x2B exp( (x)) on R d to model a perturbed density function where the optimum is distributed / exp( (x)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For example, [33] discusses training sets with 10 11 items and 10 9 distinct features, requiring novel algorithmic approaches and architectures.
Sentence 2: there has been a renewed emphasis on scaling up machine learning techniques by using massively parallel architectures; however, methods relying solely on parallelism can be expensive (both with regards to hardware requirements and energy costs) and often induce significant additional communication and data distribution overhead.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: − Upper bound on Rademacher complexity: Though existing Rademacher complexity bounds [8,22] of neural networks can be used for input data with bounded support, direct applications of those bounds to the unbounded sub-Gaussian input data yield order-level loose bounds.
Sentence 2: we develop a tighter bound on the Rademacher complexity that exploits the sub-Gaussianity of the input variables.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The data center then collects the L codeword sequences, z 1 , • • • , z L . Since all the L codewords are of the same length m, the combined data rate will be R = L × m/n.
Sentence 2: in our scenario, the data center deploys exchangeable sensors with fixed quality reproductions, ŷ1 , • • • , ŷL .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The result follows immediately.
Sentence 2: there is hope that we can reduce our our original problem of finding an optimal subgraph H ∈ H as one of finding subgraphs of H (i) ⊆ G (i) and H (j) ⊆ G (j) that are compatible, in the sense that they match up on the overlap V ij , and for which D(P G P H ) is minimized.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given a pairs of new data points ϕ n1 and ϕ n2 , we use the fact that the learned kernel is a linear transformation kernel, along with the first result of the theorem (W = αI + ΦSΦ T ) to compute the learned kernel as: As mentioned in Section 2, many Mahalanobis metric learning methods can be viewed as a special case of (2).
Sentence 2: a corollary of Theorem 1 is that we can constructively apply these metric learning methods in kernel space by solving their corresponding kernel learning problem, and then compute the learned metrics via (3).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: PROOF Suppose, we do not augment the flow in the complementary path P while augmenting P . In the next iteration the original algorithm of [22,23] picks P or any path with the same path length since the path length of P and P is equal before the iteration and they do not share any common edges.
Sentence 2: by forcing P we are not modifying the course of the original algorithm and the analysis in [22,23] holds here as well.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since the random variable ξ in Equation (3.1) can be very heavy tailed, the according elliptical distributed random vector can be heavy tailed.
Sentence 2: as has been pointed out by various authors (Tyler, 1987;Croux et al., 2002;Han and Liu, 2013b), the leading eigenvector of the sample covariance matrix Σ can be a bad estimator in estimating u 1 = Θ 1 (Σ) under the elliptical distribution.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Given such a covering, the PSOS(4) relaxation is maximize (PSOS(4)) Notice that variables ij only enter the above program if {i, j} ✓ R `for some `.
Sentence 2: the dimension of the above optimization problem is O(r P m `=1 |R `|2 ), which is O(nr) if the regions have bounded size; this will be the case in our implementation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let x k ∈ L be an example that incurs a margin violation (i.e., slack term) of ξ k . The expected contribution of x k to the SVM criterion is s(x)ξ k because x k will be drawn s(x) times on average into each re-sampled data set.
Sentence 2: training the SVM on the re-sampled data or optimizing with re-scaled slack terms lead to identical optimization problems.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Define S : Π Θ × V × (0, 1) → {0, 1} such that can be rewritten as The interpretation of L is as follows: If the ρ-quantile of H for the current policy distribution f v is greater than the constraint threshold d, we select policies in Π Θ by their H-values in order to increase the probability of drawing feasible policies.
Sentence 2: π θ is highly ranked if If the proportion of feasible policies is higher than ρ, we select policies that are both feasible and with large objective values, i.e., π θ is highly ranked if Intuitively, S can be considered as the indicator function of the highly-ranked or elite samples.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that even if ρy depends via a low-pass filter τ ρ ρy = −ρ y + Y (t) on the past spike train Y of the postsynaptic neuron, the description of the problem will turn out to be simpler if the mean firing rate is considered as a separate variable.
Sentence 2: let us write the instantaneous weight change as The goal is now to determine the simplest functionals F and G that would be consistent with the experimental protocols P 1 and P 2 introduced above.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The maximal cliques of the graphical model are the fully connected nodes (f (x 1 ), .
Sentence 2: . . , f (x n )) and the pairs (y i , f (x i )), i = 1, . . . , n. the joint probability of random variables f = {f (x i )} and y = {y i } is defined as p(f , y) for classification (1) and normalization factor Z (hereafter Z is defined such that the joint probability sums to 1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: permutation matrices A, and we have to compute probabilities for all of them.
Sentence 2: the exact filter is only applicable to problems with small N . Even if we want to keep track of K ≪ N likely permutations-as attempted by filters like the multihypothesis EKF or the particle filter-the required number of tracks K will generally have to scale exponentially with N (albeit at a slower rate).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Second, the state identity test requires solving the following problem over the function class G: The objective and the constraints are linear functionals of G, all empirical expectations involve polynomially many samples, and the number of constraints is |D h | which remains polynomial throughout the execution of the algorithm, as we will show in the sample complexity analysis.
Sentence 2: the LP oracle can directly handle this optimization problem.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In this section we give an alternative definition of bisubmodularity; it will be helpful later for describing a relationship to the roof duality.
Sentence 2: as is often done for bisubmodular functions, we will encode each half-integral value x i ∈ {0, 1, 1 2 } via two binary variables (u i , u i ) according to the following rules: 0 ↔ (0, 1) 1 ↔ (1, 0) 1 2 ↔ (0, 0) labelings in K 1/2 will be represented via labelings in the set is a set with 2n nodes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Suppose rank(X * ) ≤ k. Then we have f Proof.
Sentence 2: we have the following relation similar to (2.3): From Theorem 3.4 we know that h S+1 ≤ (1 after setting X S+2 = W S+1 , we still have h S+2 ≤ 1 e h 1 + κε 2 (according to Lemma 5.1).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: For finite graphs, the presence of cycles in the graph degrades the BP estimate and we show that the EP solution with a tree-structured approximation is less sensitive to the presence of such loops, and provides more accurate estimates for the marginal of each bit.
Sentence 2: it makes the expectation propagation with a tree-structured approximation (for short we refer to this algorithm as tree-structured EP or TEP) a more practical decoding algorithm for finite length LDPC codes.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To determine which bundles z s are sold at time t, write t = 2 j m where m is odd, and sell all bundles z s purchased during the previous 2 j−1 time steps which are not yet sold.
Sentence 2: the noise trader will sell bundles purchased at times s = t − 1, t − 2, t − 4, t − 8, . . . , t − 2 j−1 ; in particular, when t is odd we have j = 0, so no previous bundles will be sold.
Relationship: reasoning