sadat2307/MSciNLI huggingface dataset:
Domain: NeurIPS
Sentence 1: Moreover, its number of rounds is logarithmic in n and L. This is in sharp contrast with the previously best known algorithm, DISCOVER [12], where the number of rounds scales with min(m, |OP T |) 4 .
Sentence 2: fASTCOVER not only improves exponentially over DISCOVER in terms of speed but also its number of rounds decreases as the number of available machines m increases.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The proteins from these experiments form a suitable data set for this analysis, as they (a) are varied in terms of structure and "difficulty", (b) have previously unknown structures, which have been subsequently made public, (c) are timestamped and (d) they have been subject to contact prediction attempts by other groups whose results are publicly available.
Sentence 2: training on CASP9 and CASP10 data sets allowed us to avoid cross-contamination.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, both of the definitions of risk and robustness based on prediction change [32] and corrupted instance [21,20] could compute large risk and small robustness for such h. In fact, in a recent work [33] it is shown that for definitions based on corrupted input, correctness might be provably at odds with robustness in some cases.
Sentence 2: even though all these definitions could perhaps be used to approximate the risk and robustness when we do not have access to the ground truth c ′ on the new point x ′ , in this work we separate the definition of risk and robustness from how to compute/approximate them, so we will use Definition 2.1 by default.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Then We believe that some of the constants (including lower bound on d and exact upper bound on λ/σ) can be tightened, but the results demonstrate matching scaling behavior of clustering error with d, n and λ/σ.
Sentence 2: we see (ignoring constants and log terms) that The result is quite intuitive: the dependence on dimension d is as expected.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: the frequencies are large enough then X 2 is approximately χ 2 -distributed with df = (N −1)(M −1)−s degrees of freedom, where N is the number of rows, M is the number of columns, and s is the number of parameters in the H 0 model (1 for the Frank family).
Sentence 2: for a given significance level α the test accepts the hypothesis H 0 that the observed frequencies are a sample from the distribution formed by the expected frequencies, if X 2 is less than the (1 − α) point of the χ 2 -distribution with df degrees of freedom.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the widely-available sensors that provide electricity consumption information, namely the so-called "Smart Meters" that are already becoming ubiquitous, collect energy information only at the whole-home level and at a very low resolution (typically every hour or 15 minutes).
Sentence 2: energy disaggregation methods that can take this wholehome data and use it to predict individual appliance usage present an algorithmic challenge where advances can have a significant impact on large-scale energy efficiency issues.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Associated with the public key P there is a (possibly probabilistic) encryption function E P and associated with the secret or private key S there is a (deterministic) decryption function D S . Informally, the system should have the following security properties: • For any n-bit x, the value of the function E P (x) can be computed in polynomial time from inputs x and P . Similarly, D S (y) can be computed efficiently given y and S. • For any n-bit input x, D S (E P (x)) = x.
Sentence 2: decryption is the inverse of encryption.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We explicitly update the exponentially-decayed moving averages µ and σ, and optimize the rest of the model using gradient optimization, with the gradients calculated via backpropagation: These gradient equations reveal another interpretation of Batch Renormalization.
Sentence 2: because the loss is unaffected when all x i are shifted or scaled by the same amount, the functions ({x i + t}) and ({x i • (1 + t)}) are constant in t, and computing their derivatives at t = 0 gives if we consider the m-dimensional vector ∂ ∂xi (with one element per example in the minibatch), and further consider two vectors p 0 = (1, . . . , 1) and p 1 = (x 1 , .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, the guarantees obtained under these weaker assumptions usually require more intricate derivations, provide weaker noise robustness, and often do not hold uniformly for all potential target signals.
Sentence 2: to keep the exposition simple and straightforward we assume (4) which is known to hold (with high probability) for various ensembles of random matrices (e.g., Gaussian, Rademacher, partial Fourier, etc).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Note that ρ DS is a non-monotonic function of σ T . For a range of σ T corresponding to intervals shorter than the typical duration of visual fixation, ĉD αα is significantly larger than ĉS αα .
Sentence 2: fixational instability strongly influences correlated activity in the model when it moves the direction of gaze within a range of a few arcmin and is correlated over a fraction of the duration of visual fixation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Specifically, for a fixed value of tr(Σ), the quantity tr(D −1 σ )/ |||R||| F , is much smaller in in the presence of correlation.
Sentence 2: when comparing (a) with (c), and (b) with (d), we see that correlation improves the performance of our test relative to SD, as expected from the bound in Theorem 3.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: If a dynamical model is constructed based on a sequence of states x 1:N , then simulations generated by the dynamical model would be reconstructing the already observed time evolution of the system for all time steps t ≤ N , and predicting the time evolution of the system for all time steps t > N . We would like to train a dynamical model that we can ultimately use to predict the future time evolution of a given system.
Sentence 2: during training we generate the A-matrix based on only the first T /2 entries of X and Ỹ , thereby enforcing that the last T /2 entries of Ỹpred are purely predictions for how the system will evolve in time.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The full extensive game is divided into subtrees and each subtree is supplied its own abstraction: Definition 3.3 Let α 0 , α 1 , ..., α p be abstractions for the game Γ and for each i ∈ N , let G i = {G i,0 , G i,1 , ..., G i,p } be a grafting partition for player i satisfying I ∩ G i,j ∈ {∅, I} for all j ∈ {0, ..., p} and I ∈ α j,I i .
Sentence 2: each abstract information set is contained entirely in some part of the grafting partition.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since σ2 n and = σ 2 n are linear in λ and λ, we have σ2 .
Sentence 2: for the upper bound σ2 n , note that although it scales linearly with n S , the eigenvalues of K(1) scales with n, thus does not depend on π S even for moderate sizes n S . the lower bound is not as useful as the upper bound.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The contributions u i Y will thus be formed as a superposition of u i Y = u i G + u i N . Now, by Theorem 1, we know that G will be very close to zero for the latter coefficients, while on the other hand, the noise N will be equally distributed over all coefficients.
Sentence 2: the kernel PCA coefficients s = u i Y will have the shape of an evenly distributed noise floor u i N from which the coefficients u i G of the relevant information protrude (see Figure 2(b) for an example).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Similarly, if the optimal r is such that there exists j ∈ I min (r) (that is r(j) = r min ) and p j ≤ 2 −Λ , then a distribution Q that is completely concentrated on j has D(Q||P ) ≥ Λ and achieves ρ(Q) = r min as in the unrestricted game.
Sentence 2: r = r δ , and so maximizes r min .
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The weight of trip i ∈ D m,t in this prediction is given by weight where are the respective parameter values for trip i, and (l 1 , l 2 , l 3 , l 4 , l 5 ) are tunable hyperparameters.
Sentence 2: we have 6 hyperparameters to tune: (m, l 1 , l 2 , l 3 , l 4 , l 5 ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: On the other hand, the irrepresentability condition for the Lasso requires only that 2|ρ| < 1, i.e., ρ ∈ (−0.5, 0.5).
Sentence 2: in the regime |ρ| ∈ [0.2017, 0.5), the Lasso irrepresentability condition holds while our log-determinant counterpart fails.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: At the highest sampling rate (20 Hz), connections at the fastest timescales (50 ms) were estimated most effectively, whereas at the slowest sampling rates (0.2 Hz), only the slowest timescale connections (5 s) were estimated; intermediate sampling rates (2 Hz) estimated connections at intermediate timescales (0.5 s).
Sentence 2: dGC estimated robustly those connections whose process timescale was closest to the sampling rate of the data.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Setting m ρ = −σ −2 (1+A) −1 +δ and expanding (10) 2 ).
Sentence 2: as N → ∞, if the stationary points at −σ −2 (1 + A) −1 + δ exist they will define a small interval of z centred on λ u (A, σ 2 ) and so define an approximate contribution of N −1 δ(λ − λ u (A, σ 2 )) to the spectrum, in agreement with the previous calculations using replicas.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In Section 3, we formed a ranking by first independently drawing permutations π p and π q , of object sets {1, . . . , p} (vegetables) and {p + 1, . . . , n} (fruits), respectively, drawing a (p, q)-interleaving (i.e., a relative ranking permutation, τ Y ∈ Ω p,q ), and shuffling to form σ = τ Y [π p π q ].
Sentence 2: an object i ∈ {1, . . . , p} is ranked in position τ Y (π p (i)) after shuffling (and an object j ∈ {p + 1, . . . , n} is ranked in position τ Y (π q (j)) ).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: That is, update r(t) − 1 completed before the Read for g t , but update r(t) completed after.
Sentence 2: our loss (for coordinate i) is g t x r(t) , and we desire a bound on We say an update s is outstanding at time t if the Read for Update s occurs before update t, but the Update occurs after: precisely, s is outstanding at t if r(s) ≤ t < s. We let F t ≡ {s | r(s) ≤ t < s} be the set of updates outstanding at time t. We call the sum of these gradients the forward gradient sum, g fwd t ≡ s∈Ft g s . Then, ignoring constant factors and terms independent of T , we show that AdaptiveRevision has a per-coordinate bound of the form Theorem 3 gives the precise result as well as the n-dimensional version.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: However, it is generally impractical to simply initialize the model to include all possible features: exact inference in such a model is almost invariably intractable, and approximate inference methods such as loopy belief propagation [17] are likely to give highly inaccurate estimates of the gradient, leading to poorly learned models.
Sentence 2: we propose an algorithm schema that gradually introduces features into the model, and lets the L 1 -regularization scheme eliminate them via the optimization process.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: More formally, the view classifier G(E(x i t ); θg) → p t maps an encoding at timestep t to a probability distribution over possible views V . Learning with GRL is adversarial in that θ g is optimized to increase G's ability to discriminate encodings from different views, while GRL reverses the sign of the gradient that flows back to E, which results in the encoder parameters θ e learning representations that reduces the view classification accuracy.
Sentence 2: essentially, we minimize the cross-entropy loss for the view classification task with respect to θ g , while maximize it with respect to θ e . we define the view classification loss as the sum of the cross-entropy loss for the entire sequence: where i is the ground-truth view of the input.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: By setting the other models in relation to the best joint model (cICA:= 100%), we are able to tell apart the relative contributions of bandpass filtering (HAD= 91%), particular filter shapes (SYM= 93%, ICA= 94%), contrast gain control (cHAD= 98.6%) as well as combined models (cSYM= 99%, cICA := 100%) to redundancy reduction (see Table 1).
Sentence 2: orientation selectivity (ICA) contributes less to the overall redundancy reduction than any model with contrast gain control (cHAD, cSYM, cICA).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: is a monoid with identity element £ ; ¡ distributes over ; and ¢ is an annihilator for ¡ : for all X `Y T ¤ %X 0¡ ¢ 38 ¢ a¡ bX 58 ¢ .
Sentence 2: a semiring is a ring that may lack negation.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: In words, Proposition 1 implies that under the stated global-optimality preserving conditions, if any element of E converges to zero during optimization with an arbitrary descent algorithm, it will remain anchored at zero until the end.
Sentence 2: if the algorithm prematurely errs in setting the wrong element to zero, meaning the wrong support pattern has been inferred at any time during an optimization trajectory, it is impossible to ever recover, a problem naturally side-stepped by MC where the support is effectively known.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Consider an agent situated in a partially observable domain: It executes an action; the action may change the state of the world; this change is reflected, in turn, by the agent's sensors; the action may have some associated cost, and the new state may have some associated reward or penalty.
Sentence 2: the agent's interaction with this environment is characterized by a sequence of action-observation-reward steps, known as instances [7].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We aim to learn a jigsaw such that, given an image set, pieces of the jigsaw image satisfy the following criteria: • each piece is similar in appearance and shape to several regions of the training images; • any of the training images can be approximately reconstructed using only pieces from the jigsaw (a piece may be used more than once in a single image); • pieces are as large as possible for a particular accuracy of reconstruction.
Sentence 2: while allowing the jigsaw pieces to have arbitrary shape, we ensure that such pieces are shared across the entire image set, exhaustively explain the input image set, and are also large enough to be discriminative.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: So, E is approximately independent of G\E.
Sentence 2: given v, r, and denoting by λ 1 = (a + (k − 1)b)/k and λ 2 = (a − b)/k the two eigvenvalues of P Q in the symmetric case, the expected number of intra-community neighbors at depth r from v is approximately 1 k (λ r 1 + (k − 1)λ r 2 ), whereas the expected number of extra-community neighbors at depth r from v is approximately 1 k (λ r 1 − λ r 2 ) for each of the other (k − 1) communities.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We denote by d : Y ×Y ×H → [0, 1] the distortion function, which measures the dissimilarity among two elements of the output space Y and one element of the latent space H. (Note that the distortion function is general in the sense that the latent element may not be used in some applications.)
Sentence 2: the goal is to find a w ∈ W that minimizes the decoder distortion, that is: In the above equation, the angle brackets indicate that we are inserting a pair (ŷ, ĥ) = f w (x) into the distortion function.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Moreover, the model has a translational invariance in the skills and a scale invariance in β/σ 0 and τ /σ 0 .
Sentence 2: we fixed µ 0 = 1200, σ 0 = 400 and computed the log-evidence L := P (y|β, τ ) for varying values of β and τ (see Figure 2 (right)).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: ( 4) is as follows: a feature y is to be associated with a cluster c in proportion to a weighted, though flattened, geometric mean of the "W-projected centroids" p(y|c,w), priored by p(c).
Sentence 2: 2 This scheme overweighs y's that contribute to c evenly across W. clusters satisfying Eq.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We now study how pooling changes the requirements for efficient decoding by the target neuron.
Sentence 2: mathematically speaking, pooling maps the vector r onto a scalar ρ equal to the sum of the individual activities: ρ = r k . the mutual information that can be extracted by any decoder that only pools it inputs is given by the following expression: where P (ρ|s) and P (ρ) are the the stimulus-conditional and stimulus-unconditional probability of observing a pooled population response ρ on a single trial.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Using soft assignment, we define the soft quantization of z as where we write the centers as a matrix , where e(z) is the center in C nearest to z.
Sentence 2: we can now write: Now, instead of computing Ẑ via hard nearest neighbor assignments, we can approximate it with a smooth relaxation Z := C[φ(z (1) ), • • • , φ(z (m) )] by using the soft assignments instead of the hard assignments.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Therefore, in our scenario, the data center deploys exchangeable sensors with fixed quality reproductions, ŷ1 , • • • , ŷL .
Sentence 2: lastly, the tth symbol of the estimate, T , is to be calculated by majority vote [3], overall performance of the system can be measured by the expected bit error frequency for decisions by the majority vote (3), In this paper, we consider two limit cases of decentralization levels; (1) The extreme situation of L → ∞, and (2) the case of L = R. The former case means that the data rate for an individual sensor information vanishes, while the latter case results in the transmission without coding techniques.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Let the feature correlation matrix be R = (R st ) ∈ p×p , clearly, R = R T , R st represents the correlations between features s and t, i.e., To let the selected features uncorrelated as much as possible, for any two features s, t, if their correlations R st > θ, then we put them in an exclusive group.
Sentence 2: only one feature can survive.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since one major contribution of this paper is learning a classifier for structured objects that is defined over the complete part space P, we now examine the differences of semi-supervised and transductive learning in more detail.
Sentence 2: the most common approach to realize the smoothness assumption is to construct a data dependent kernel k S derived from the graph Laplacian on a nearest neighbor graph on the labeled and unlabeled input patterns in the sample S. k S is not defined on observations that are out of the sample.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: As in the previous example, it is easy to check that L t (h, g|Z t 1 1 ) = |c c ⇤ | 2 t 2 .
Sentence 2: in this case, one can show that (7) reduces to 1 3 (T + 1)(2T + 1)|c c ⇤ | 2 and therefore, if Periodic signals.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: The former will influence the choice of the default ν, the latter the form of constraints P. When the number of labeled and unlabeled examples is sufficiently large, it is reasonable to assume that the empirical distribution π(x) over all examples (labeled and unlabeled) is a faithful representation of π(x).
Sentence 2: we consider defaults with ν(x) = π(x), shown to work well in species distribution modeling [13].
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Since S n is bilinear in p and q, the plug-in estimator for S n is unbiased.
Sentence 2: that is, our estimator for p, q H s is Here, we present our main theoretical results, bounding the bias, variance, and mean squared error of our estimator for finite n. By construction, our estimator satisfies via (6) and Cauchy-Schwarz, the bias of the estimator Ŝn satisfies p − p n H s is the error of approximating p by an order-Z n trigonometric polynomial, a classic problem in approximation theory, for which Theorem 2.2 of [15] shows: In combination with (8), this implies the following bound on the bias of our estimator: Theorem 1.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Unfortunately, some ideas in Bassily and Smith [4] such as Johnson-Lindenstrauss lemma do not scale to population sizes of millions of users.
Sentence 2: in order to have a smooth trade-off between accuracy and communication cost (as well as the ability to protect privacy in repeated data collection, which will be introduced in Section 3) we introduce a modified version of Duchi et al.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: To derive the theoretical guarantee, we first re-formulate (4).
Sentence 2: in this model, we can rewrite by assuming M = V T V , and using the fact that kM k ⇤ = min V :M =V T V kV k 2 F , the Factorization HOI (4) can be converted to the following nuclear norm regularization problem: where f w,M (e + , e ) := w T (e + e ) + (e + ) T M e + (e ) T M e . We then derive the guarantee for the following equivalent hard-constraint form: where ⌦ is the set of observed group comparisons (there can be repeated pairs in ⌦).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: From the final log-product term, it can be seen (from Schur's determinant identity [5] and the definition of σ t,b (x t,b )) that the product of the last B − 1 terms is exactly the B − 1 principal minor of K t,1 formed by the indices corresponding to S = {x t,b } B b=2 .
Sentence 2: it is straightforward to see that the UCB-PE algorithm is really just (B − 1)-DPP maximization via the greedy algorithm.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: 2(a) shows an example evolution of Q x4 for k = 1.
Sentence 2: to further save computation time, we exploit two observations: First, if A is a subset of a connected component Q ∈ Q S , adding hyperedge S (A) to Q S will not change Q S . we do not test any hyper-edge A which is contained in a connected component.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: We choose g(z) as the point where this line intersects with the affine hyperplane 1 T ẑ = 1 containing the probability simplex.
Sentence 2: g(z) is set equal to αz + (1 − α)q, where α = 1+Kq i zi+Kq (we denote it as α(z) as α is a function of z).
Relationship: reasoning
Domain: NeurIPS
Sentence 1: This assumption constrains the timevarying knowledge state: it can make at most one transition from 0 to 1 over the sequence of trials.
Sentence 2: the {K t } can be replaced by a single latent variable, T , that denotes the trial following which a transition is made, leading to the BKT generative model: where λ L is the probability that a student has mastered the skill prior to performing the first exercise, λ M is the transition probability from the not-mastered to mastered state, λ G is the probability of correctly guessing the answer prior to skill mastery, and λ S is the probability of answering incorrectly due to a slip following skill mastery.
Relationship: reasoning
Domain: NeurIPS
Sentence 1: Learning techniques can be used to choose the similarity measure, which could be adaptive and learned from a large number of examples of classes not involved in the test.
Sentence 2: given a large number of training images of a large number of objects labeled with their actual classes, and provided two pictures of unknown objects (objects which do not appear in the training pictures), we want to decide if these two objects are actually the same object.
Relationship: reasoning