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[EQUATION] Thus, the final seed-length is [MATH] . The theorem follows. 7.5 Proof of Claim 7.14 Claim With probability at least [MATH] [MATH] is a [MATH] -ROBP and [MATH] |
Proof. The second part follows from Claim 7.1 . We are left to prove the first part. Let [MATH] be the pseudorandom restriction, where [MATH] and [MATH] Assume there are [MATH] colliding layers in [MATH] and let [MATH] be their indices. For [MATH] , call a layer [MATH] “good” under the choice of [MATH] if [MATH] and th... |
For [MATH] let [MATH] be the event that none of layers [MATH] is good. Recall that [MATH] is sampled from a [MATH] -biased distribution with marginals [MATH] , and |
[MATH] is sampled from a [MATH] -biased distribution. For [MATH] to happen, we must have a partition [MATH] such that all layers [MATH] for [MATH] are in [MATH] and all layers [MATH] for [MATH] are in [MATH] but the edged marked by [MATH] in the [MATH] -th layer do not collide. For any fixed [MATH] and fixed partition ... |
[EQUATION] (using [MATH] ). Overall, [EQUATION] assuming [MATH] is a sufficiently small constant and [MATH] for a sufficiently large constant [MATH] By the union bound, |
[EQUATION] Under the event that all [MATH] are false, we get that [MATH] can be written as [MATH] where each [MATH] is a width- [MATH] ROBP with at most [MATH] colliding layers and at most [MATH] vertices on the first and last layer. |
7.6 Pseudorandom generator for unordered 3ROBPs In this section, using the recent generator of Chattopadhyay, Hatami, Hosseini, Lovett CHHL18 and a Fourier bound by Steinke, Vadhan and Wan SVW17 we show that we can also handle unordered 3ROBPs, thus proving Theorem |
Lemma 7.16 (Lemma 3.14 SVW17 Let [MATH] and let [MATH] be a width- [MATH] ROBP with at most [MATH] colliding layers. Then, for all [MATH] it holds that [MATH] |
Theorem 7.17 (Theorem 4.5 CHHL18 Let [MATH] be a family of [MATH] -variate Boolean functions closed under restrictions. Assume that for all [MATH] for all [MATH] [MATH] Then, for any [MATH] , there exists a log-space explicit PRG which fools [MATH] with error [MATH] , whose seed length is |
[MATH] Corollary 7.18 There is a log-space explicit PRG that [MATH] -fools unordered ROBPs with width [MATH] , length [MATH] and at most [MATH] colliding layers using seed length |
[EQUATION] Proof of Theorem The proof is essentially the same as that of Theorem , where instead of using the generator from Theorem 7.9 to set the bits after the pseudorandom restrictions, we use the generator from the above corollary. The final seed-length has a worse dependence on [MATH] as we need to set [MATH] in ... |
7.7 Pseudorandom generator for locally-monotone width-3 ROBPs In this section, we construct pseudorandom generators that [MATH] -fool unordered locally-monotone 3ROBPs with seed-length [MATH] Our dependency on [MATH] is much better than in Section , and we get nearly logarithmic (in [MATH] ) seed-length even for error ... |
We remark that read-once CNFs and read-once DNFs are special cases of locally-monotone 3ROBPs, hence our result extends the result of GMRTV GMR 12 that constructs an [MATH] -PRG for the former classes of functions using seed length [MATH] |
Theorem 7 For any [MATH] , there exists a log-space explicit PRG that [MATH] -fools unordered locally-monotone 3ROBPs with seed-length [MATH] |
Proof. The pseudorandom generator samples a string in [MATH] as follows: 1. Apply the pseudorandom restriction [MATH] from Claim 7.1 with [MATH] and [MATH] |
2. Assign the coordinates in [MATH] using the pseudorandom generator [MATH] with [MATH] It is clear that the seed-length is [MATH] by Claims 7.1 and 6.8 |
Next, we show that the generator [MATH] -fools unordered locally monotone 3ROBPs. By Claim 7.14 , with probability at least [MATH] [MATH] is a [MATH] -ROBP for [MATH] . Furthermore, [MATH] Now, observe that since [MATH] is locally monotone, for each layer [MATH] either [MATH] or [MATH] has at most [MATH] end-vertices (... |
Thm. 4.2 states that whenever [MATH] simplified, it can be written as [MATH] where [MATH] are subprograms of [MATH] , and [MATH] By Claim 6.7 , the distribution [MATH] fools [MATH] with error at most [MATH] , for all [MATH] Thus, it fools [MATH] with error at most [MATH] Overall, we get |
[EQUATION] Acknowledgements We would like to thank Oded Goldreich and Salil Vadhan for very helpful comments on an earlier version of this manuscript. |
Appendix A Appendices A.1 Proof of Theorem 3.10 In this section, we view the Boolean functions computed by branching programs as functions [MATH] . For any set [MATH] , this changes the sum [MATH] by a factor of [MATH] , which we can afford. |
Let [MATH] be a ROBP of length [MATH] and width [MATH] . Recall that [MATH] denote the layers of vertices in [MATH] For a vertex [MATH] in the branching program we denote by [MATH] the sub-branching program ending in the [MATH] -th layer and having [MATH] the only accepting state. We denote by [MATH] the sub-branching ... |
[EQUATION] The main technical result from CHRT17 is the following theorem: Theorem A.1 CHRT17 , Thm. 2] Let [MATH] be an ordered read-once, oblivious branching program of length [MATH] and width [MATH] . Then, |
[EQUATION] We are ready to prove a corollary of this theorem, namely Theorem 3.10 Theorem A.2 (Thm. 3.10 , restated) Let [MATH] be a width- [MATH] length- [MATH] ROBP. Let [MATH] [MATH] [MATH] , and [MATH] be a [MATH] -biased distribution over [MATH] with marginals [MATH] , where [MATH] Then, with probability at least ... |
[EQUATION] Claim A.3 For all [MATH] , the following holds with probability at least [MATH] over [MATH] : for all [MATH] and [MATH] and [MATH] |
[EQUATION] Proof. Fix [MATH] and [MATH] . Letting [MATH] denote the branching program [MATH] we get [MATH] from Theorem A.1 Thus, |
[EQUATION] Finally, we conclude by applying the Markov inequality and a union bound, as there is a total of at most [MATH] branching programs [MATH] and at most [MATH] choices for [MATH] |
Theorem 3.10 follows from the next claim which uses Claim A.3 with [MATH] and [MATH] that ensure [MATH] and [MATH] Indeed, with probability at least [MATH] , the spectral-norm of [MATH] is at most [MATH] |
Claim A.4 Suppose that [MATH] is such that the events in Claim A.3 hold for [MATH] such that [MATH] Then for every [MATH] such that [MATH] |
[EQUATION] Proof. We prove by induction on [MATH] that Eq. ( 15 ) holds for all [MATH] , for any [MATH] and [MATH] . Note that [MATH] itself is of the form [MATH] for [MATH] being the accept node in the final layer (w.l.o.g. there exists only one such node). The case [MATH] is handled by Claim A.3 , since [MATH] For [M... |
[EQUATION] This completes the induction, and hence the claim follows. A.2 Restatement of XOR-lemma for functions fooled by small-biased spaces |
In this section we show how Lemma 5.1 is a restatement of Thm 4.1 in GMR 12 . We recall the following equivalence between having sandwiching approximations with small spectral-norm and being fooled by every small-biased distribution. |
Lemma A.5 DETT10 Let [MATH] be a function. Then, the following hold for every [MATH] If [MATH] has [MATH] -sandwiching approximations of spectral-norm at most [MATH] , then for every [MATH] -biased distribution [MATH] on [MATH] [MATH] |
If for every [MATH] -biased distribution [MATH] on [MATH] [MATH] , then [MATH] has [MATH] -sandwiching approximations of spectral-norm at most [MATH] |
We recall GMR 12 , Thm. 4.1] Theorem A.6 GMR 12 , Thm. 4.1] Let [MATH] be functions on disjoint input variables such that each [MATH] has [MATH] -sandwiching approximation of spectral-norm at most [MATH] Let [MATH] be a multilinear function in its inputs. Let [MATH] be defined as [MATH] . Then [MATH] has [MATH] -sandwi... |
We translate the domain [MATH] to [MATH] to get a restatement of the previous theorem. Theorem A.7 GMR 12 , Thm. 4.1] [MATH] -version) |
Let [MATH] be functions on disjoint input variables such that each [MATH] has [MATH] -sandwiching approximation of spectral-norm at most [MATH] Let [MATH] be a multilinear function in its inputs. Let [MATH] be defined as [MATH] . Then [MATH] has [MATH] -sandwiching approximations of spectral-norm at most [MATH] |
Proof. We take [MATH] to be [MATH] respectively. We get that [MATH] has [MATH] -sandwiching approximations of spectral-norm at most [MATH] , for all [MATH] We take [MATH] to be |
[MATH] Since [MATH] is multilinear, so is [MATH] By Theorem A.6 , we get that [MATH] has [MATH] -sandwiching approximations of spectral-norm at most [MATH] Since [MATH] |
we got that [MATH] as a [MATH] -sandwiching approximations of spectral-norm at most [MATH] Finally, we restate Lemma 5.1 and prove it. |
Lemma A.8 Let [MATH] Let [MATH] be functions on disjoint input variables such that each [MATH] is [MATH] -fooled by any [MATH] -biased distribution. Let [MATH] be a multilinear function in its inputs. Then [MATH] is [MATH] -fooled by any [MATH] -biased distribution. |
Proof. Using the second item in Lemma A.5 , since [MATH] are [MATH] -fooled by any [MATH] -biased distribution, we have that there exist [MATH] -sandwiching approximations of spectral-norm at most [MATH] Thus by Thm. A.7 [MATH] has [MATH] -sandwiching approximations of spectral-norm at most [MATH] Set [MATH] and [MATH]... |
A.3 Pseudorandom restrictions for the composition of 3ROBPs We restate and prove Lemma 7.13 Lemma A.9 Let [MATH] be 3ROBPs on disjoint sets of variables of [MATH] Let [MATH] be any Boolean function. Then, [MATH] is [MATH] -fooled by the pseudorandom partial assignment in Claim 7.1 with parameter [MATH] |
Proof. Claim 7.1 applies Theorem iteratively [MATH] times with error parameter [MATH] . Thus, it suffices to show that under each application of pseudorandom restriction from Theorem the acceptance probability of [MATH] changes by at most [MATH] |
Let [MATH] Let [MATH] be the sets of variables on which [MATH] depend. We write [MATH] in the Fourier basis: [MATH] Thus, [MATH] Recall that the pseudorandom assignment in Theorem is composed of two stages: Let [MATH] and [MATH] |
(c) Assign the coordinates in [MATH] using a [MATH] -biased distribution [MATH] Recall that for a fixed [MATH] , the bias-function of any program [MATH] behaves the same under any relabeling of the layers in [MATH] . We imagine as if these layers are relabeled so that a collision is possible, and denote this relabeled ... |
[MATH] By Theorems 4.1 and 4.2 , with probability at least [MATH] the choice of [MATH] and [MATH] , we can write each [MATH] for [MATH] as a linear combination of [MATH] where the sum of coefficients in absolute value is at most the number of variables in [MATH] (i.e., [MATH] ), and each [MATH] is a ROBP on at most [MA... |
[EQUATION] By Theorem , each XOR of [MATH] -length width-3 ROBPs is [MATH] -fooled by the pseudorandom assignment defined by Step 3 above, thus the overall error is at most [MATH] |
# Source: arxiv 1806.04398 # Title: Attentive cross-modal paratope prediction # Sections: all # Downloaded: 2026-03-03T05:15:52.674860+00:00 |
Attentive cross-modal paratope prediction Abstract Antibodies are a critical part of the immune system, having the function of directly neutralising or tagging undesirable objects (the antigens) for future destruction. Being able to predict which amino acids belong to the paratope , the region on the antibody which bin... |
Machine Learning, ICML Introduction Antibodies are Y-shaped proteins used by the immune system to neutralise pathogens such as bacteria and viruses. This is done when the antibody binds to the unique molecules on the pathogen called antigens. With antibodies being the most important class of biopharmaceuticals, knowing... |
Traditional attempts for predicting the binding amino acids (the paratope ) were based on hard coded physical models, requiring vast amounts of information. Predictors such as Antibody i-Patch (Krawczyk et al., 2013 use as input the full structural information of the antibody and the antigen, while proABC (Olimpieri et... |
Only recently, Parapred (Liberis et al., 2018 —a hybrid architecture consisting of convolutional and recurrent layers—has become the state of the art technique. However, its usage of recurrent layers represents a performance bottleneck, and it discards the information about the target antigen entirely. |
In this work, we outperform Parapred by addressing its limitations and leveraging the bleeding-edge techniques in the language modelling community, such as à trous convolutions (Kalchbrenner et al., 2016 and self-attention (Vaswani et al., 2017 , while also significantly lowering computation time. We then manage to fur... |
Dataset and Preprocessing We used a subset of the Structural Antibody Database (SAbDab) (Dunbar et al., 2014 , which provides crystal structures of antibody-antigen complexes, in order to train and evaluate our models. The subset was chosen under the same criteria as in Liberis et al. ( 2018 |
1. Antibodies having variable domains of their heavy ( [MATH] ) and light ( [MATH] ) chains; 2. Structure resolution better than 3Å; |
3. No two antibody sequences have [MATH] 95% sequence identity; 4. Each antibody has at least five amino acid residues in contact with the target antigen. |
The paratope is contained within the complementarity determining regions CDRs ) of the antibody. We identify the CDRs within the sequence of each antibody using the Chothia numbering scheme (Al-Lazikani et al., 1997 , and use each CDR as an independent training sequence. |
For each residue in the CDR, we use the following features to obtain its feature vector, [MATH] A one-hot encoding of the amino acid type (20 possible types + 1 additional for an unknown type); |
A one-hot encoding of the chain ID of the CDR (6 possible types—three on the heavy chain (H1, H2, H3) and three on the light chain (L1, L2, L3). Consequentially, all residues within the same CDR will receive the same encoding; |
Seven additional features, summarised by Meiler et al. ( 2001 , representing physical, chemical and structural properties of the given amino acid type (may be seen as a fixed embedding ). |
In addition, for all of the complexes in the dataset, the antigens were proteins . This allowed us to also extract the 1D residue sequences on the antigen. There is no equivalent for CDRs on the antigen, so the entire antigen sequence is extracted, and each residue’s feature vector [MATH] is obtained exactly as for [MA... |
Methods 3.1 Antibody-only We build up on the developments of Parapred by substituting its recurrent layers with a combination of à trous convolutional layers (for efficient modelling of longer-range dependencies) and a self-attentional layer (allowing for efficiently covering the sequence). We will refer to this archit... |
A high-level layout of the architecture is presented in Figure 1. It receives as input the vector of antibody residue features [MATH] , and consists of: |
A stack of three à trous (dilated) convolutional layers: 1. 64 features, kernel size 3, dilation rate 1; 2. 128 features, kernel size 3, dilation rate 2; |
3. 256 features, kernel size 3, dilation rate 4. Then, a self-attention mechanism is applied on the computed intermediate features, [MATH] |
Lastly, a pointwise fully-connected (dense) layer is applied to classify each considered antibody amino acid as binding or non-binding. |
All à trous convolutional layers and the self-attention layer employ the exponential linear unit (ELU) (Clevert et al., 2015 activation function, while the prediction layer uses the logistic sigmoid function to perform binary classification. All the layers are initialised using Xavier initialisation (Glorot & Bengio, 2... |
The leveraged self-attention mechanism (depicted in Figure 2) is the same as the one utilised by Veličković et al. ( 2018 . Taking a set of intermediate antibody residue features [MATH] , a shared neural network is applied to all pairs of residues, producing attention coefficients |
[EQUATION] indicating the importance of residue [MATH] ’s features to residue [MATH] . Here the neural network [MATH] is a single-layer feedforward neural network, parametrised by a weight vector [MATH] , and applying the LeakyReLU nonlinearity (with negative input slope [MATH] ): |
[EQUATION] where [MATH] represents transposition and [MATH] is the concatenation operation. Here, [MATH] is a shared, learnable linear transformation of the residue features (preserving their dimensionality at 256)—adding further expressivity to the layer. |
Once computed, the attention coefficients are normalised using the softmax function, for easy comparability across different residues: |
[EQUATION] Lastly, using the normalised attention coefficients, we compute a linear combination of all antibody residues’ features for each attending antibody residue: |
[EQUATION] which represents the final output of the layer (summarised by Figure 3 (left)). The regularisation methods used in this architecture are: |
-regularisation (with [MATH] ); Dropout (Srivastava et al., 2014 (with [MATH] on the final layer and [MATH] on all the other ones); |
Batch normalisation (Ioffe & Szegedy, 2015 on the output of each layer; A skip connection (He et al., 2015 over self-attention, to preserve positional information of the residues. |
The model (as well as all subsequent models) is trained using the Adam SGD optimiser—with base learning rate of 0.01 and other hyperparameters as presented in Kingma & Ba ( 2014 —for 20 epochs with a batch size of 32. |
3.2 Antibody-Antigen With similar motivation as before, we extract features from antibody and antigen amino acid residues by applying, independently to both, a stack of three à trous convolutional layers (with exactly the same hyperparameters as for the antibody-only model). The self-attention in the antibody-only para... |
We will focus on describing our cross-modal attentional layer here, as the other layers are defined exactly the same as in Fast-Parapred (with identical hyperparameters). The input to the layer is a set of antibody residue features |
[MATH] , a set of antigen residue features [MATH] and for each antibody residue [MATH] a set [MATH] which marks the antigen residues which are in a fixed-range neighbourhood from [MATH] . This neighbourhood was chosen to restrict the number of antigen residues being attended over by any antibody residue to 150. The att... |
[EQUATION] Using the normalised attention coefficients, we then compute a linear combination of the corresponding antigen residues in the neighbourhood, for each attending antibody: |
[EQUATION] conveniently summarised by Figure 3 (right). The result is, in a similar way to the Antibody-only method, passed through a pointwise convolutional layer and a logistic sigmoid non-linearity is applied, in order to classify each considered antibody amino acid residue as binding or non-binding. |
We apply the same regularisation as for the antibody-only model—along with a skip connection over the cross-modal attention (which is in this case critical , as the layer entirely discards antibody features). |
Results 4.1 Quantitative Results We perform ten runs of 10-fold crossvalidation (with 10 distinct splits of the data into 10 folds) on Parapred, Fast-Parapred and AG-Fast-Parapred. For each, we monitor ROC-AUC, Matthews correlation coefficient (which we also report for proABC ; Table 1), the wall-clock time it takes to... |
Fast-Parapred has achieved the state-of-the-art-level result on antibody-only paratope prediction, while requiring only half the computational time of Parapred; |
AG-Fast-Parapred has significantly outperformed this result, for the first time successfully leveraging antigen information in a deep paratope predictor, while relying solely on convolutional and attentional layers, removing the dependency on recurrent layers entirely. |
It should be noted that AG-Fast-Parapred still improves on the epoch time of Parapred, despite working with input sizes that are up to [MATH] larger. |
4.2 Qualitative Results We visualise, using PyMOL, the computed binding probabilities of AG-Fast-Parapred , on a test antibody-antigen complex, in Figure (left)—revealing that its neural network has learnt to appropriately infer positional information (predicting higher probabilities for the residues closer to the anti... |
# Source: arxiv 1806.04399 # Title: A new (2+1) dimensional integrable evolution equation for an ion acoustic wave in a magnetized plasma # Sections: all # Downloaded: 2026-03-02T08:55:24.233851+00:00 |
A new (2+1) dimensional integrable evolution equation for an ion acoustic wave in a magnetized plasma Abstract A new, completely integrable, two dimensional evolution equation is derived for an ion acoustic wave propagating in a magnetized, collisionless plasma. The equation is a multidimensional generalization of a mo... |
Introduction Active research on nonlinear phenomena in plasma physics has grown extensively and gained much importance over the past few decades due to failure of linear theory in explaining phenomena related to large amplitude waves, wave- particle, wave-wave interactions etc Chen However, the complexity of the associ... |
This equation is also investigated extensively in various areas of plasma systems like dusty plasma Amin Moslem , multicomponent plasma Sabry in explaining rogue waves Moslem Langmuir hydromagnetic , relativistic laser plasma interactions Javan |
and various other fields. Peregrine soliton of NLS equation which is used to describe rogue waves is experimentally observed in a multicomponent plasma with negative ions Bailung . A complex Ginzberg-Landau equation is derived in compressional dispersive Alfvenic waves in a collisional magnetoplasma hydromagnetic |
which reduces to standard NLS equation in a collisionless plasma. The discussed equations are all (1+1) dimensional, but in practical circumstances the waves observed in laboratory and space are certainly not bounded in one dimension. Franz Franz et.al have shown that a purely 1D model cannot account for the observed f... |
KP which has also been used in various branches of plasma such as inhomogeneous plasma with finite temperature drifting ions Dahiya , ultracold quantum magnetospheric plasma Mushtaq , electron positron ion plasma Shahmansouri and also in other areas. The stability of their solutions under transverse perturbations was a... |
which is more isotropic in transverse direction was first derived for describing weakly nonlinear ion acoustic waves in strongly magnetized lossless plasma in 2D Zakharov2 . It was also reported that this equation is not integrable under inverse scattering method Bhimsen Infeld and till date only three polynomial conse... |
Mace Wazwaz . A 2D generalization of NLS equation is DS equation which was also derived for electrostatic ion waves Nishinari , electron acoustic wave Ghosh , space and laboratory dusty plasma Annou , and in cylindrical geometry Xue . For special choice of coefficients DS equation converges to DS1 equation which is ana... |
In this work we have derived a completely integrable 2D nonlinear evolution equation in lossless magnetized plasma with asymmetric scaling on transverse variable. This equation involving only local interactions of dependent variables was derived earlier in hydrodynamic system MyRogue The 2D generalizations of NLS equat... |
The paper is organized as follows. The derivation of the (2+1) dimensional, integrable, evolution equation for an ion acoustic wave in a magnetized plasma is given in section II with asymmetric scaling on transverse variables. Nonlinear frequency correction and modulation instability of the evolution equation are discu... |
II Derivation of two dimensional integrable equation for electrostatic waves propagating in a magnetized plasma A new two dimensional integrable evolution equation for the propagation of nonlinear waves in magnetized plasma is derived in this section. We consider the propagation of electrostatic waves in a magnetized p... |
[EQUATION] where [MATH] are electron, ion densities and ion fluid velocity, magnetic field and electrostatic potential respectively and [MATH] is a dimensionless parameter given by [MATH] . For convenience, we have used the following normalization resulting in dimensionless parameters: electron and ion densities normal... |
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