Generating semi-factual and counterfactual explanations from images requires methods for extracting and adjusting appropriate image features. This short paper presents initial research on a counterfactual generation method for images based on class-to-class variational autoencoders (C2C-VAEs). Initial experiments illustrate substantial speed increase in counterfactual generation while suggesting that the method achieves reasonable counterfactual quality compared to the state of the art. The paper closes by discussing tradeofs of the approach.
the diference patterns between two classes. Given a standard VAE that can extract a feature
vector from a case, a C2C-VAE can embed the diference between two feature vectors into a
diference embedding or reconstruct a feature diference from a diference embedding. Previously
used to generate creative samples from limited data [3], a C2C-VAE can modify a source case
into a target case , where the line connecting and in the embedding space possesses desirable
characteristics for generating counterfactuals: (
Desiderata for Counterfactuals: For a given query of class , a counterfactual is a case of a diferent class , and a semi-factual is a case of the class . Keane and Smyth [4] propose three criteria for good counterfactuals: A good counterfactual avoids prolixity (it is minimally diferent from the query ), achieves sparsity (it difers from the query in the fewest features) and has plausibility (The counterfactual is realistic for the domain). The PIECE Approach to Counterfactual Generation: Given a query Image , a GAN generator and a CNN classifier to be explained , PIECE first performs GAN inversion, locating a latent vector using gradient descent such that () = . PIECE then modifies the penultimate layer output of CNN = () to ′ by identifying exceptional features according to the weight vector of the last layer. The counterfactual output (′) is generated by optimizing ′ such that the MSE loss between ′ and ((′)) is minimized.
The C2C-VAE Approach: A C2C-VAE [3] is a type of variational autoencoder that learns an embedding space of the diference pattern between features of two classes [ 3]. Because C2C-VAE works with case features, it requires a feature extractor for domains in which pre-extracted features are not available (e.g. images) and, if new case generation is desired, a procedure ′ to reconstruct a case from a feature vector. Both the feature extractor and case constructor can be implemented by the encoder and decoder of a standard variational autoencoder.
Given a pair of cases and , their features are () and (), and the feature diference Δ(, ) = () − (). C2C-VAE encodes Δ using an encoder function as (< Δ, , >) and decodes this embedding using a decoder function ′ as Δ′ = ′((< Δ, , >). With its encoder and decoder, C2C-VAE can sample a new feature diference embedding (< Δ, , >) from a normal distribution and construct the corresponding feature diference Δ = ′((< Δ, , >). C2C-VAE can also synthesize a case of class by adapting a source case as = ′( () − Δ) (See Figure 1).
VAE encoder f encoder f case s case t reconstruction loss + Regularizer (gaussian) f(s) f(t) decoder f' recovered sample f'(f(t)) fΔ = f(s) - f(t)
C2C-VAE Conditional Parameter
Cs + Ct encoder g feature difference fΔ reconstruction loss + Regularizer (gaussian) difference encoding g(<fΔ, Cs,Ct>)
decoder g' recovered difference g'(g(<fΔ, Cs,Ct>),<Cs,Ct>) 3. Using C2C-VAE to Generate Counterfactuals: In previous work applying C2C-VAE to creative case generation [3], we noticed that the generated case often preserves the visual characteristics of . We hypothesized that this is due to the encoder-decoder pair of C2C-VAE, and ′, learning to recognize the feature diferences in Δ(, ) = () − () more related to class change to , therefore ignoring other less related features. This relates the sparsity needed for a good counterfactual, as shown in Figure 2.
Following the core design of C2C-VAE in Section 2, given a query of the class , C2C-VAE can be used to generate a guide of another class . A counterfactual of the class can be found on the interpolation between () and () such that: is near the boundary of and (avoiding prolixity); is generated following an average diference Δ(, ) pattern (sparsity, explained in the previous paragraph); And is interpolated from () and () within the embedding space of a standard VAE, thus conforming to the training data distribution (plausibility). Diversity of can be introduced by perturbing either or Δ(, ). 3.1. The CVC Algorithm to Generate Counterfactuals The CVC counterfactual generation algorithm is based on C2C-VAE and the Native Guide technique [5]. CVC has two steps: retrieval and adaptation.
Retrieval: CVC first randomly samples random vectors 1 . . . from a normal distribution with mean 0 and standard deviation (e.g. = 1) in the feature diference embedding (a) Retrieval Process of CVC (b) Adaptation Process of CVC space of C2C-VAE. Given query , its class and a target class , CVC uses C2C-VAE to generate K feature diferences Δ′1 . . . Δ′ by decoding 1 . . . using equation 1.
Δ′ = ′(< , , >)
CVC then generates K guide candidate cases 1, ..., such that = ′( () − Δ′), and selects as the guide case that minimizes mean squared error (, ) in pixel space. Figure 3a shows an example of the retrieval step. In contrast to the guide feature ′ of PIECE, the CVC guide can be directly visualized, therefore providing more explanability.
Adaptation: After selecting guide and its corresponding VAE feature () = () − Δ′, CVC interpolates between () and () in the VAE’s latent space to construct counterfactuals (or semi-factuals ), as shown in equation 2.
= ′((1 − ) * () + * ()), 0 ≤ ≤ 1
In equation 2, is a variable that determines the relative weight of interpolation between query
and guide . A small value (<0.5) means the output is more similar to , and a large value
(>0.5) means the output is more similar to . As shown in figure 3b, CVC is able to synthesize
meaningful results for diferent values. The diferent values of allow CVC to find and
that are closest to the decision boundary. For qualitative evaluation, we found that = 0.5 was
suitable to visualize the diference between query, counterfactual and guide in our test domain.
(
We performed an ablation study comparing adaptation results using (
We compared CVC to PIECE for the incorrect classifications test-set from Kenny and Keane
[2]. We evaluated PIECE under two settings: (
As observed by Kenny and Keane, it is dificult to quantitatively assess the counterfactual desiderata for image data. We measured both eficiency and proximity: • Inference time: Time used to generate each image. • SSIM and PSNR: We calculate Structural Similarity Index (SSIM) and Peak Signal to Noise Ratio (PSNR) between query and generated counterfactual image to measure their proximity.
We proposed a novel counterfactual generation algorithm (CVC) that significantly reduces computational time comparing to the current state-of-the-art. Initial qualitative results suggest 1https://drive.google.com/drive/folders/1bZ_oy7eFt7LubmVSXmrOIIenO2yKwlW9?usp=sharing comparable quality counterfactuals. As a benefit inherited from C2C-VAE, CVC is applicable to one-shot learning settings (however this is not illustrated in this study). In addition, CVC does not require weight vectors of any layers of the CNN classifier to be explained, so is applicable to any black-box classifier.
This paper presents initial work. Additional evaluation is needed on other data sets and for image quality. The limited image reconstruction quality of the “vanilla" VAE structure we are currently using may prevent CVC from generating plausible results on higher resolution and more realistic images, so our future work will also involve testing other image synthesis models such as GANs. Another direction is to apply CVC to counterfactual generation for tabular data.
This work was funded by the Department of the Navy, Ofice of Naval Research (Award N0001419-1-2655).