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Recovering Weak Signals with Normalizing Flows
Sarod Yatawatta
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
This paper proposes a method to recover weak scientific signals obscured by dominant nuisance signals using normalizing flow models. The approach leverages the statistical invariance of target signals and uses the Jacobian of the calibration mapping (derived from the influence function) to reconstruct the original signal distribution from calibration residuals. The method is validated through linear regression simulations and radio interferometric data models.
Entities (9)
Relation Signals (5)
Normalizing Flows → usedfor → Signal Recovery
confidence 95% · we propose the use of normalizing flow models to reconstruct calibration-affected weak signals.
Calibration → causes → Signal Distortion
confidence 90% · this calibration process inherently distorts or partially suppresses the underlying signal of interest.
Jacobian → derivedfrom → Influence Function
confidence 85% · The statistical relationship between the input data and the output residual can be described by using the influence function... This relationship enables the closed form calculation of the Jacobian
Radio Interferometry → exampleof → Calibration
confidence 80% · The statistical performance of calibration (in radio interferometry) has been studied
Cosmology → uses → Weak Signal Recovery
confidence 80% · A case in point are the signals that originate from the Universe that feed several research areas in cosmology.
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Abstract
Abstract:In many scientific disciplines, weak signals of interest are obscured by dominant nuisance signals that are several orders of magnitude stronger. Recovering these weak signals requires subtracting the dominant ones; however, this calibration process inherently distorts or partially suppresses the underlying signal of interest. To address this problem, we propose the use of normalizing flow models to reconstruct calibration-affected weak signals. By leveraging the statistical invariance of the target signals and assuming minimal initial suppression, our framework effectively recovers the lost signal components. We provide a comprehensive theoretical overview of this normalizing flow-based recovery method and demonstrate its efficacy using simulated data.
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- Source: https://arxiv.org/abs/2609.06382v1
- Canonical: https://arxiv.org/abs/2609.06382v1
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Recovering Weak Signals with Normalizing Flows Preprint: APS/X Sarod Yatawatta Email: yatawatta@astron.nl Affiliation: ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD, Dwingeloo, The Netherlands. September 6, 2026 Abstract In many scientific disciplines, weak signals of interest are obscured by dominant nuisance signals that are several orders of magnitude stronger. Recovering these weak signals requires subtracting the dominant ones; however, this calibration process inherently distorts or partially suppresses the underlying signal of interest. To address this problem, we propose the use of normalizing flow models to reconstruct calibration-affected weak signals. By leveraging the statistical invariance of the target signals and assuming minimal initial suppression, our framework effectively recovers the lost signal components. We provide a comprehensive theoretical overview of this normalizing flow-based recovery method and demonstrate its efficacy using simulated data. Keywords: Deep learning, Generative models, Cosmology, Radio telescopes, Cosmic microwave background I Introduction Many disciplines in physics rely on the accumulation of observational data and the recovery of weak, hidden signals in that data. A case in point are the signals that originate from the Universe that feed several research areas in cosmology. Such signals have statistical stability as an advantage but they are overwhelmed by nuisance signals that are of many orders of magnitudes higher in power level (e.g., Galactic and extra-Galactic foregrounds). The estimation and the removal of nuisance signals from the observed data are performed by many specialized techniques unique to each scientific discipline but in this paper, we broadly consider them to be a form of calibration. In calibration, a parameterized model for the nuisance signals are constructed and subtracted from the data (see Fig. 1). The desired weak signals are assumed to remain intact in the residual. Due to the statistical stability of the weak, hidden signals, it is generally assumed that the residual data can be accumulated without any limit to reach the desired dynamic range for their detection. The statistical performance of calibration (in radio interferometry) has been studied both theoretically [1, 2, e.g.,] and empirically [3, 4, 5, e.g.,]. Its effect on the residual and on the weak signals hidden in the residual is well known. However, no method exists that can recover the original weak signals hidden in the residual. This work intends to improve on that. Figure 1: A schematic of the signal flow. The nuisance signals are subtracted from the observed data x by calibration to get the residual y. In this paper, we propose the use of normalizing flows [6] to recover the undistorted weak signals in the residual after calibration. Normalizing flows have been used in many data processing applications such as cosmological parameter estimation and model construction [7, 8, 9, 10, 11, e.g.,] and simulations [12, 13, 14, 15, e.g.,]. There are many more potential application areas [16, 17] and our work covers one such application. The statistical relationship between the input data and the output residual can be described by using the influence function [18, 19, 20, 1]. This relationship enables the closed form calculation of the Jacobian of the mapping between the input and the output residuals. We incorporate this into a normalizing flow model to train the base distribution which enables the determination of the statistics (mainly first order) of the hidden weak signals. Existing work based on flow models mostly learn the Jacobian (or its inverse) as part of the training process. In contrast, we work with a flow model where the Jacobian of the mapping is externally provided and hence we are not constrained to a bijective mapping. Calibration is a computationally demanding task in data processing and this is performed by specialized and highly optimized software to cope with large data volumes. As a bi-product of calibration, the Jacobian of the mapping from data to residual is also produced. In contrast, common applications of normalizing flows [21, e.g., contractive residual flows] need to learn it themselves. Parameter estimation in (ill-constrained) linear models has some similarity to this work. Prior distributions are trained using flow models in [22] while stochastic differential equations for structured noise are combined with parameter estimation in [23]. Contrastive flows for signal and background separation (in event classification) are studied in [24] where the background signal is considered a nuisance parameter. A separate flow model for nuisance parameters is trained in [25] for interventional (causal) density estimation. Similarly, in [26], factorized flow models are used to directly estimate the distribution of interest as well as the nuisance parameters. Note that in our method, the nuisance parameters are handled by calibration, without having to train a flow model for computational efficiency. Normalizing flows that flow in both directions (from data to noise and from noise to data) are considered in [27] that uses learnt approximate inverse mappings for heteroscedastic noise modeling in image processing. Flow models with non-invertible Jacobians are considered in [28] where the flow is embedding the data onto a low dimensional manifold. In this paper, we propose a method that uses a normalizing flow for density estimation where the Jacobian of the forward mapping is explicitly provided. Existing work use normalizing flows for profiling systematic uncertainties in simulation-based inference or act as explicit priors for inverse problems. In this work, we focus on the structural degradation of the signal of interest downstream by calibration. We consider the signal in the traditional sense to be a nuisance (Fig. 1) and the true signal of interest to be hidden in the noise. We do not have the freedom to select the nuisance model (that depends on the physics of the observational setup). The nuisance model is generally considered to be non-linear and non-invertible. A major assumption of our method is that the signal hidden in the noise is weak compared to the nuisance signal and the noise is independent of the nuisance signal. Moreover, we assume statistically stable noise compared to the nuisance signal that may have variations temporally, spatially and spectrally (for example, we can consider the noise to be of cosmological origin while the nuisance to be of Galactic and instrumental origin). Under these assumptions (that are reasonable for experiments measuring signals of cosmological origin), we illustrate a method to recover the weak signals hidden in the noise in this paper. The remainder of this paper is organized is as follows: in section I, we provide a theoretical overview of calibration and the statistical relationship between the input and the output residual. In section I, we provide an overview of normalizing flows and the setup we use to train the flow model. Next in section IV, we provide results using a linear regression example as well as using a non-linear example (radio interferometric calibration) before drawing our conclusions in section V. Notation: Lowercase bold letters refer to column vectors (e.g., y). Uppercase bold letters refer to matrices (e.g., C). Unless otherwise stated, all variables are real numbers. The set of complex numbers is given as ℂC and the set of real numbers is given as ℝR. The matrix inverse, pseudo-inverse, transpose, and Hermitian transpose are referred to as (⋅)−1(·)^-1, (⋅)†(·) , (⋅)T(·)^T, and (⋅)H(·)^H, respectively. The matrix Kronecker product is given by ⊗ . The identity matrix is given by I (size depends on the context). The Frobenius norm (for a matrix) or the L-22 norm (for a vector) is given by ∥⋅∥\|·\| and the L-11 norm is given by ∥⋅∥1\|·\|_1. The vectorization of a matrix is represented by vec(⋅)vec(·). I Data Model We consider the data flow illustrated in Fig. 1. The observed data are denoted as x (∈ℝD ^D), which consists of a parametric model () s( θ) ((⋅) s(·) is a mapping from ℝMR^M to ℝDR^D) and additive noise n (∈ℝD ^D). The unknown parameters θ (∈ℝM ^M) parameterize the nuisance signal for example by modeling the foreground signals and instrumental systematics. The observed data is given by (1) =()+. x= s( θ)+ n. (1) During calibration, the systematics model parameters θ are estimated, for example by using maximum likelihood estimation or any similar technique to get θ ^=argminf(,) θ= θ f( x; θ) (2) where f(,)f( x; θ) is the negative likelihood or similar function that is minimized to estimate θ. Note that the noise n is assumed to have a known distribution (for example zero mean white Gaussian noise) during solving (2). This is only possible because the weak signal hidden in n is hardly noticeable compared to the power of the nuisance signal s()s( θ), i.e., ‖()‖≫E\| s( θ)\| E\ n\. Due to the same reason, in order to extract the weak signal of scientific interest in the noise, we need to find the residual y, =−(^). y= x- s( θ). (3) The hope is that the statistics of the residual ∼pY() y p_Y( y) are similar to the statistics of the noise ∼p() n p( n) and therefore, we are ably to study the weak signals of interest. However, in reality, this is not the case. Consider the mapping from the data x to the residual y =() y= T( x) (4) where (⋅) T(·) is a mapping from from ℝDR^D to ℝDR^D and if ∼pY() y p_Y( y) and ∼pX() x p_X( x), we have pX()=||pY()p_X( x)=| J|\ \ p_Y( y) (5) where J is the Jacobian of the mapping (⋅) T(·), i.e., =∂()∂T J= ∂ T( x)∂ x^T (∈ℝD×D ^D× D) and ||| J| is the absolute value of its determinant. From (1), we also see that p()=pX()p( n)=p_X( x) (6) because the noise is independent of () s( θ) and ∂()∂T= ∂ s( θ)∂ n^T= 0. From (5) and (6) we see that it is possible to recover p()p( n) and extract the weak signals hidden therein (we focus exclusively on EE\ n\). Before proceeding further, we state the assumptions: • The nuisance signal is dominant compared to the weak signal hidden in noise, i.e., ‖()‖≫E\| s( θ)\| E\ n\. • The residual y is close to the noise n, i.e., ‖()−(^)‖≈0|| s( θ)- s( θ)||≈ 0. • At the solution of (2), a local minimum of the cost function exists, i.e., ∂f(,)∂T= ∂ f( x; θ)∂ θ^T= 0 • We have the ability accumulate multiple observations where in each observation the parameters θ change, creating diversity. For example, in radio interferometry, the rotation of the Earth as well as the variation of systematic errors over time and frequency creates this condition. • The base distribution p()p( n) remains statistically stable in contrast to () s( θ) for multiple observations. In radio interferometry, spatial variation is compensated by a phase shift in Fourier space. This (complex) phase shifting matrix is diagonal with complex exponentials in the diagonal, making its determinant unity. Hence for this particular case, any phase shifting does not affect the form of p()p( n). Under the aforementioned assumptions, it has been shown by many [29, e.g.,] that (,)=+∂()∂T(∂2f(,)∂∂T)†∂2f(,)∂∂T J( x, θ)= I+ ∂ s( θ)∂ θ^T ( ∂^2f( x; θ)∂ θ∂ θ^T ) ∂^2f( x; θ)∂ θ∂ x^T (7) evaluated at θ or at the local minimum of f(,)f( x; θ). The proof of (7) is given in appendix A. Note also that (7) is in fact the influence function [18, 20]. We see that the residual statistics pY()p_Y( y) are not the statistics of the weak signal we seek. Furthermore, the Jacobian (,^) J( x, θ) (∈ℝD×D ^D× D) is not necessarily full rank of D (can lose rank at most by the number of degrees of freedom used by θ, i.e., M) and hence not necessarily invertible. The conundrum we face is extracting the statistics of pX()p_X( x) or pN()p_N( n) from the residual y and the a-priori Jacobian (,^) J( x, θ). In section I, we will explore the use of normalizing flows for this purpose. I Normalizing flows Normalizing flows provide a mechanism to train a deep neural network (DNN) to model probability density functions undergoing a transform [30, 31, 6]. In our case (see Fig. 1), we consider the noise density p()p( n) undergoing a transformation due to calibration to reach the density of the residual, pY()p_Y( y). We consider the base distribution pX()p_X( x) or p(,)p( n; ψ) to be parameterized by ψ. From density transform, we have the log likelihood of y logpY()=logpX()−log|(,^)| p_Y( y)= p_X( x)- | J( x, θ)| (8) ∝logp(,)−log|(,^)|. p( n; $ψ$)- | J( x, θ)|. The eigenvalues of the Jacobian (,^) J( x, θ) are used to efficiently evaluate log|(,^)| | J( x, θ)|, i.e., the logarithm of the absolute value of the determinant |(,^)|| J( x, θ)|. Direct maximization of (8) for likelihood maximization is ill constrained, mainly because () T( x) in (5) is not necessarily invertible. Normalizing flows with non-invertible Jacobians have been used with additional constraints for improved stability [28]. Following the same procedure, we introduce an additional constraint such that the residual y is close to n; in other words the correlation T n^T y is high. With this additional constraint, we formulate the loss function to maximize as =argmaxlogp(,)−log|(,^)| ψ= ψ \ \ p( n; ψ)- | J( x, θ)| (9) subjecttoT>γ \ to\ \ n^T y>γ where γ∈ℝ+γ ^+ is the lower bound for the expected correlation between n and y. We use an inequality constraint in (9) instead of a penalty term or an equality constraint because we do not have a measure of the exact correlation between n and y. With inequality constraints, we use the augmented Lagrangian method [32] where we introduce g(,)=△([γ−T‖¯‖¯]+)2g( n, y) 0.0pt= ( [γ- n^T y \| n\|\ \| y\| ]_+ )^2 (10) making g(⋅,⋅)g(·,·) active only when positive to satisfy the constraint, where [⋅]+[·]_+ is the ReLU() operation. Note that we find the normalized inner product between n and y where ‖¯ \| n\| and ‖¯ \| y\| denote the norms of n and y averaged over the minibatch. In this manner, we overcome the need to adjust γ too much depending on the problem. We consider stochastic training where we have a minibatch ℬB to sample data. With (10), we have the augmented Lagrangian for minibatch ℬB L()= L( $ψ$)= (11) E,(,^)∼ℬ,∼p(,)[−logp(;)+log|(,^)| E_ y, J( x, θ) , n p( n; $ψ$) [- p( n; $ψ$)+ | J( x, θ)| . +ρ2g(,)2+zg(,)] .+ ρ2g( n, y)^2+z\ g( n, y) ] where ρ∈ℝ+ρ ^+ is the regularization factor and z∈ℝz is the Lagrange multiplier. After several updates of ψ, we update the Lagrange multiplier z as z←z+ρg(,).z← z+ρ\ g( n, y). (12) In certain situations, pre-training the model with the loss l()=E∼ℬ,∼p(,)[‖−‖2]l( $ψ$)=E_ y , n p( n; $ψ$) [\| n- y\|^2 ] (13) will be helpful before training the normalizing flow. In Algorithm 1, we present the pseudocode for training the normalizing flow model. More practical details such as using warmup and learning rate schedulers are presented in Appendix D. In section IV, we present results using two examples. Algorithm 1 Train normalizing flow 1: Epochs e, batch size b, cadence C, γ,ρ 2: Initialize z←0z← 0, ψ 3: if pre training is enabled then 4: for Epoch in e do 5: for Batch ℬB do 6: Update ψ by minimizing (13) 7: end for 8: end for 9: end if 10: ni←1n_i← 1 11: for Epoch in e do 12: for Batch ℬB do 13: Update ψ by minimizing (11) 14: if nin_i is a multiple of C then 15: Update z using (12) 16: end if 17: ni←ni+1n_i← n_i+1 18: end for 19: end for 20: Return solution ψ IV Simulation Results We provide two examples where we seek a weak signal hidden in the noise. We first provide a simple linear example and next a complicated example based on radio interferometry. Note that the nuisance models () s( θ) in (1) are vastly different in the two examples but we can apply the same method as summarized in Algorithm 1 for both. IV.1 Linear model We consider the nuisance model to be linear, i.e., =+ x= A $θ$+ n (14) where A (∈ℝD×M ^D× M) is the design matrix that is known for each observation of x (∈ℝD ^D) and θ (∈ℝM ^M) are the parameters to estimate. For each observation, both the design matrix and the parameters θ will change while the base density of n remain fixed. We consider using elastic net regression for estimation of θ as described in Appendix B, and the residual is calculated as =−^. y= x- A $θ$. (15) The calculation of (,) J( x, θ) is also described in Appendix B. It may look plausible to combine multiple observations of x for estimating θ to get an improved result. However, this is not possible because for each observation, we have a unique unknown value of θ. In order to illustrate, we run a simulation with D=15D=15 and M=10M=10. We generate 9000090000 samples of x with the following settings applied to each sample: • We generate A with entries drawn from (0,1)N(0,1) and θ with entries drawn from (0,1)U(0,1). • We generate the noise n with samples drawn from the multivariate Gaussian (,)N( μ, I) where μ is a constant vector (initialized with entries drawn from (0,0.1)N(0,0.1)) for all simulations (note that ‖≈0\| μ\|≈ 0). We add the noise to the signal A θ by scaling the signal such that the signal to noise ratio is 1. • We solve the elastic net regression problem to find the solution θ as in (23). Using this solution, we find the residual y as in (15). • The input to the DNN is y as well as metadata A and θ. The output is ~ n, sampled from p(,)p( n; $ψ$). An example simulation is shown in Fig. 2. Note that the observed data x and the residual data y have higher power than the weak signal μ that is hidden in noise. Even the residual y has higher power than μ, highlighting the need for averaging multiple observations of y to reach the weak signal. Figure 2: An example simulation where the observed data x, the residual data y and the ground truth weak signal μ are shown. We train our normalizing flow using the 9000090000 samples and further details are given in Appendix D. For evaluating the trained model, we simulate 30003000 additional samples using the same strategy as above. We average the residuals y of all 30003000 samples to find the estimates for μ. For each simulation, we also do a prediction using our trained flow model which is also averaged. The results are shown in Fig. 3. We see that while the averaged residual gives a suppressed result as the estimate of μ, the averaged prediction using the trained normalized flow gives a result more in agreement with the ground truth. Figure 3: Estimated weak signal using 30003000 samples compared with the ground truth (note the change in scale compared to Fig. 2). The averaged residual y gives a lower estimate of μ (suppressing the signal and losing power). The average prediction by the trained normalized flow model ~ n gives a more accurate result. IV.2 Radio interferometric model We present the data model used by a radio interferometer in this section. The observed data for baseline pqpq at frequency ν and at any given time instance are given by pqν V_pqν (∈ℂ2×2 ^2× 2) which is expressed as [33] pqν=∑k∈[1,K]pkνpqkνqkνH+pqν. V_pqν= _k∈[1,K] J_pkν C_pqkν J_qkν^H+ N_pqν. (16) The baseline pqpq is formed by correlating the voltages received by stations p and q. We consider the total number of stations (or receivers) to be N. The nuisance signal in radio interferometry is generated by the sources in the sky (both Galactic and extra-Galactic). We model them as the summation of K discrete signal as given by the first term on the right of (16). Each signal in this summation, say the k-th, is a composition of the true signal from the sky, i.e., pqkν C_pqkν (∈ℂ2×2 ^2× 2) and terms representing systematic errors pkν,qkν J_pkν, J_qkν (∈ℂ2×2 ^2× 2) of receivers p and q along the path of the k-th source. The true source signal pqkν C_pqkν depends on the intensity of the source as well as its position in the sky (lk,mk,nkl_k,m_k,n_k in radians) and the baseline coordinates in Fourier space (upq,vpq,wpqu_pq,v_pq,w_pq in wavelengths). The weak signal of interest is in the noise pqν N_pqν (∈ℂ2×2 ^2× 2) in (16). The noise is assumed to be having a complex circular Gaussian density with zero mean and σ2σ^2 I as covariance. However, due the presence of the unknown weak signal of interest, the true noise statistics are slightly different from the assumed statistics. With calibration we estimate the systematic errors pkν J_pkν,pqkν C_pqkν for all p,q,k and ν (also with time dependence when data are taken over multiple sample instances). Thereafter, the residual pqν R_pqν (∈ℂ2×2 ^2× 2) is extracted as pqν=pqν−∑k∈[1,K]^pkνpqkν^qkνH. R_pqν= V_pqν- _k∈[1,K] J_pkν C_pqkν J_qkν^H. (17) where ^pkν J_pkν and ^qkν J_qkν represent the solutions obtained by calibration for the systematic errors. We can rewrite both (16) and (17) in the form of (1) and (3), respectively, by stacking the complex data as a vector of real components for all p,q and ν and by considering the parameterization of pkν J_pkν as a vector of real numbers in θ. In Appendix C we provide further details on calculating the influence function and (7) for the radio interferometic data model. For this example, we set a simulation of a radio interferometric array with N=14N=14 receivers, each receiver similar in instrumental response to a LOFAR high-band station [34]. We generate 6000060000 snapshot observations (with duration of one time sample with 1010 s integration time and a bandwidth of 1 channel with 200200 kHz width) pointing at random locations in the sky as training data. • The beam pointing direction is randomly chosen and the epoch of observation is chosen so that the pointing direction is above the horizon. • The nuisance signal is modeled as clusters of sources at K=6K=6 distinct directions in the sky, at random separations from the pointing center. Their flux densities are randomly to be in [0.1,200]U[0.1,200] Jy/PSF. • The systematic errors pkν J_pkν are generated to have entries drawn from a complex circular Gaussian distribution with zero mean and unit variance. In addition, an I is added to the diagonal of pkν J_pkν for all p, k and ν. • The frequency of the simulation is randomly chosen from [110,150]U[110,150] MHz. • The noise pq N_pq in (16) is modeled by first drawing samples from a complex circular Gaussian distribution with zero mean and unit variance. To simulate the hidden weak signal, we generate a diffuse sky model that is kept fixed for all simulations and is centered at the north celestial pole. The diffuse sky model is simulated using a shapelet model (with random coefficients) as described in [35]. Both these components are added together to create pq N_pq. Note that the diffuse sky model has lower power compared to the noise component. Finally, the observed data pqν V_pqν in (16) are created by adding pq N_pq to the nuisance signal. The nuisance signal is scaled such that the nuisance signal to noise ratio is kept at 1010. Note that this is done so that the noise distribution p()p( n) remain stable over all simulations. • Direction dependent calibration is performed using the simulated data using specialized calibration software to get the residual (17), the same software also calculates the eigenvalues of the influence function (26) at no significant additional cost. Note that we assume perfect knowledge of the K=6K=6 sky model (nuisance signal) but no information of the diffuse sky is used during calibration. • For training the DNN, we use the residual pqν R_pqν (17) as input (in vectorized form as y). For N=14N=14 we have D=N(N−1)/2×8=728D=N(N-1)/2× 8=728 values as input. We use the following as metadata: The sky model of the nuisance signal of K=6K=6 source clusters. The calibration solutions ^pkν J_pkν for all p and k. The observed data pqν V_pqν for all p and q. Finally, the coordinates of the baselines in Fourier space, i.e., all upq,vpq,wpqu_pq,v_pq,w_pq triplets for all p and q. • The output of the DNN is a prediction of the noise signal ^pqν N_pqν with the weak diffuse emission hopefully intact. In Appendix D, we provide further practical details about the DNN model and the training strategy. After training the DNN, we evaluate its performance by generating a separate test observation. We generate an observation with 22 hr duration (1010 s integration time), with 2020 frequencies equally spaced in the range [110,150][110,150] MHz. The remaining details of the simulation are similar to the settings used in the generation of training data as described previously. In Fig. 4, we show images made using the test observation. Data at all frequencies are imaged separately and average to get the final image. The calibration is performed per each time and frequency sample and the DNN model prediction is also done at this resolution. One noteworthy aspect in the DNN model as opposed to the DNN model trained in the linear example is the use of Monte Carlo dropout [36]. Therefore, multiple predictions are performed at each time and frequency sample and their average is taken as the final result. Data Ground truth Residual Prediction Figure 4: Average images made using the test observation (all frequencies), covering about 8×88× 8 square degrees in the sky, centered at the north celestial pole. Note that the beam pointing direction of this observation is elsewhere in the sky while the weak (diffuse) signal is simulated to be at the pole. The four columns from left to right show the images made using the observed data (prior to calibration), the ground truth (without noise), the residual (after calibration) and the prediction from the DNN, respectively. Note the difference in the color scale in the left column, where the dominant component is the nuisance signal and its artifacts. The four rows from top to bottom correspond to the Stokes I, Q, U and V polarizations. Note that the images made using the residual has significant suppression of the desired signal while the DNN prediction closely agrees with the ground truth. The ground truth images are without any noise while the DNN prediction contains the expected noise as well. This is evidenced by the Stokes V images in the last row, where the image made by uncalibrated data (leftmost column) and the DNN prediction (rightmost column) have more or less the same power level. From the images in Fig. 4 we clearly see that the DNN prediction does a more accurate prediction of the hidden signal compared to just using the residual, closely matching the ground truth. Note that the ground truth images do not show any noise (because only the diffuse sky model is simulated). V Conclusions We have presented the use of normalizing flows for recovering weak signals that are affected by calibration. Using simulated data for training and testing the normalizing flow model, we have shown the improved reconstruction of the weak signal. Note that with real observations, we do not need any simulated data for training the DNN as we can directly feed the observed data and metadata into the training process. Specifically for radio interferometry, by using this method, we can combine various observations done aimed at diverse science goals and even observations done by various radio telescopes to probe deeper into the Universe. Data Availability Statement: All data used in this paper are simulated and the software used for simulations and evaluations are provided at https://github.com/SarodYatawatta/InfluenceFlow. Appendix A Proof of (7) We give the proof based on prior work [37, 38, 29]. Let xmx_m be the m-th element of x, we take the derivative of (3) ∂xm=∂xm−∂xms()|=^. ∂ y∂ x_m= ∂ x∂ x_m- ∂ x_ms( θ) _ θ= θ. (18) Using the chain rule (at = θ= θ) ∂xm()|=^=∂()∂T|=^×∂^∂xm ∂ x_m s( $θ$) _ $θ$= $θ$= ∂ s( $θ$)∂ $θ$^T _ $θ$= $θ$× ∂ $θ$∂ x_m (19) where ∂()∂T∈ℝD×M ∂ s( $θ$)∂ $θ$^T ^D× M and ∂xm∈ℝM×1 ∂ $θ$∂ x_m ^M× 1. At the solution of (2), we have a local minimum, i.e., ∂f(,)∂|=^= ∂ f( x, $θ$)∂ $θ$ _ $θ$= $θ$= 0. Let f′(,)=∂f(,)∂f ( x, $θ$)= ∂ f( x, $θ$)∂ $θ$ for simplified notation. Then f′(,^)=f ( x, $θ$)= 0 and taking derivative of both sides with respect to xmx_m, we get ∂f′∂xm∂xm∂xm+∂f′∂^∂^∂xm= ∂ f ∂ x_m ∂ x_m∂ x_m+ ∂ f ∂ $θ$ ∂ $θ$∂ x_m= 0. Simplifying this leads to (20), ∂^∂xm=−(fθθ(,))−1fxmθ(,)|= ∂ $θ$∂ x_m=- (f_θ( x, $θ$) )^-1f_x_mθ( x, $θ$) _ $θ$= $θ$ (20) where fθθ(,)=△∂2f(,)∂∂T∈ℝM×M, f_θ( x, $θ$) 0.0pt= ∂^2f( x, $θ$)∂ $θ$∂ $θ$^T ^M× M, (21) fxmθ(,)=△∂2f(,)∂xm∂∈ℝM×1. f_x_mθ( x, $θ$) 0.0pt= ∂^2f( x, $θ$)∂ x_m∂ $θ$ ^M× 1. (22) Taking the derivative of (3) with respect to x, we get I for the first term and for the second term, we use (19) and (20) above to get (7). Appendix B Linear model With elastic net regression, the estimate $θ$ is obtained as ^=argmin(‖−‖2+ρ2‖2+ρ1‖1) $θ$= $θ$ (\| x- A $θ$\|^2+ _2\| $θ$\|^2+ _1\| $θ$\|_1 ) (23) where ρ1 _1 and ρ2 _2 are regularization factors (both set at 0.0010.001). The influence function is obtained as + I+A where we use (7) to get =12(T+(ρ2+ρ1δ(‖)))−1(−2T)A= A 12 ( A^T A+( _2+ _1δ(\| $θ$\|)) I )^-1 (-2 A^T ) (24) where δ(⋅)δ(·) is the Dirac delta function. Appendix C Radio interferometric model In order to simplify the notation, we introduce the canonical selection matrix p A_p (∈ℝ2×2N ^2× 2N), where only the p-th block is ∈ℝ2×2 I ^2× 2, the remaining entries are 00. With this notation, we have for example, pkν=pkν J_pkν= A_p J_kν where kν J_kν (∈ℂ2N×2 ^2N× 2) is a block matrix collecting systematic errors for all N stations along the k-th direction at frequency ν into one. We minimize the cost function gν(kν,k∈[1,K])= g_ν( J_kν;k∈[1,K])= (25) ∑p,q‖pqν−∑k∈[1,K]pkνpqkν(qkν)H‖2 _p,q\| V_pqν- _k∈[1,K] A_p J_kν C_pqkν( A_q J_kν)^H\|^2 to find the solutions kν J_kν for all k and ν, using data at all baselines (all possible p and q, which is N(N−1)/2N(N-1)/2 for one frequency). Consider xp′q′rx_p q r to be an element in the observed data vector x corresponding to the baseline formed by stations p′p and q′q , and r∈[0,7]r∈[0,7] is the index to denote which component of the 2×22× 2 matrix it represents. We take the derivative of (17) by xp′q′rx_p q r to get vec(∂pqν∂xp′q′r) ( ∂ R_pqν∂ x_p q r ) =vec(∂pqν∂xp′q′r) =vec ( ∂ V_pqν∂ x_p q r ) (26) −∑k∈[1,K] - _k∈[1,K] (pqkν^kνHqT)T⊗pvec(∂kν∂xp′q′r) ( C_pqkν J_kν^H A_q^T )^T A_pvec ( ∂ J_kν∂ x_p q r ) where analogous to (20), we have [39] vec(∂kν∂xp′q′r) ( ∂ J_kν∂ x_p q r ) (27) =(gradk(gν(k′ν,k′∈[1,K])))−1 = (D_ J grad_k(g_ν( J_k ν;k ∈[1,K])) )^-1 ×(q′kνp′q′kνH)T⊗p′Tvec(∂p′q′ν∂xp′q′r) × ( A_q J_kν C_p q kν^H )^T A_p ^Tvec ( ∂ V_p q ν∂ x_p q r ) and gradk(gν(k′ν,k′∈[1,K]))= _ J grad_k(g_ν( J_k ν;k ∈[1,K]))= (28) ∑p,q(−(pqkνH)T⊗pTpqνq−pqkνT⊗qTpqνHp _p,q (-( C_pqkν^H)^T A_p^T R_pqν A_q- C_pqkν^T A_q^T R_pqν^H A_p . + + (pqkνkνHqTqkνpqkνH)T⊗pTp .( C_pqkν J_kν^H A_q^T A_q J_kν C_pqkν^H)^T A_p^T A_p . + + OPEN(pqkνHkνHpTpkνpqkν)T⊗qTq). .( C_pqkν^H J_kν^H A_p^T A_p J_kν C_pqkν)^T A_q^T A_q ). Finally, we rearrange (26) in vectorized form, with real and imaginary parts of each ℂ2×2C^2× 2 matrix represented by 8 real values in the vector x for any given p,q and ν. Adding spectral regularization as in [40], modifies the above formulae as in [1] but we leave them out here for simplicity. Appendix D Deep learning model and training details In Fig. 5 we provide a rough schematic of the DNN used in the examples in section IV. All layers in the DNN use dense linear layers with sigmoid linear unit (SiLU) activation (but no activation in the last layer). The dimensions of each linear layer in Fig. 5 are shown. We use D to denote the input dimension, H to denote the hidden dimension and Mx to denote the metadata dimension, respectively. If the metadata dimension is higher than H, H is replaced by the metadata dimension for that particular flow (until concatenation). Depending on the example, there may be different number of metadata types (22 for the linear model and 44 for the radio interferometric model). We can also change the depth of the DNN in Fig. 5 by changing the parameter T (indicating T linear blocks are appended sequentially, with SiLU activation). Optionally, we introduce dropout at two locations in the DNN (with dropout rate set at 0.1). A residual connection is also created between the input data and the output after concatenation. Figure 5: A schematic of the DNN used in both examples discussed in section IV. The exact parameters for D, H, Mx, and T vary for each example. We use the AdamW optimizer [41] for training the DNN models. We use learning rate scheduling with a warmup period and cosine decay thereafter. Additional details of the training are given in Table 1. 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