Does My Multimodal Model Learn Cross-modal Interactions? It’s Harder to Tell than You Might Think!

Modeling expressive cross-modal interactions seems crucial in multimodal tasks, such as visual question answering. However, sometimes high-performing black-box algorithms turn out to be mostly exploiting unimodal signals in the data. We propose a new diagnostic tool, empirical multimodally-additive function projection (EMAP), for isolating whether or not cross-modal interactions improve performance for a given model on a given task. This function projection modifies model predictions so that cross-modal interactions are eliminated, isolating the additive, unimodal structure. For seven image+text classification tasks (on each of which we set new state-of-the-art benchmarks), we find that, in many cases, removing cross-modal interactions results in little to no performance degradation. Surprisingly, this holds even when expressive models, with capacity to consider interactions, otherwise outperform less expressive models; thus, performance improvements, even when present, often cannot be attributed to consideration of cross-modal feature interactions. We hence recommend that researchers in multimodal machine learning report the performance not only of unimodal baselines, but also the EMAP of their best-performing model.


Introduction
Given the presumed importance of reasoning across modalities in multimodal machine learning tasks, we should evaluate a model's ability to leverage cross-modal interactions. But such evaluation is not straightforward; for example, an early Visual Question-Answering (VQA) challenge was later "broken" by a high-performing method that ignored the image entirely (Jabri et al., 2016).
One response is to create multimodal-reasoning datasets that are specifically and cleverly balanced to resist language-only or visual-only models; examples are VQA 2.0 (Goyal et al., 2017)

Empirical Multimodally Additive Projection
Figure 1: We introduce EMAP, a diagnostic for classifiers that take in textual and visual inputs. Given a (black-box) trained model, EMAP computes the predictions of an image/text ensemble that best approximates the full model predictions via empirical function projection. Although the projected predictions lose visual-textual interaction signals exploited by the original model, they often perform suprisingly well. (Suhr et al., 2019), and GQA (Hudson and Manning, 2019). However, a balancing approach not always desirable. For example, if image+text data is collected from an online social network (such as for popularity prediction or sentiment analysis), post-hoc rebalancing may obscure trends in the original data-generating processs. So, what alternative diagnostic tools are available for better understanding what models learn?
The main tool utilized by prior work is model comparison. In addition to comparing against text-only and image-only baselines, often, two multimodal models with differing representational capacity (e.g., a cross-modal attentional neural network vs. a linear model) are trained and their performance compared. The argument commonly made is that if model A, with greater expressive capacity, outperforms model B, then the performance differences can be at least partially attributed to that increased expressivity.
But is that a reliable argument? Model perfor-mance comparisons are an opaque tool for analysis, especially for deep neural networks: performance differences versus baselines, frequently small in magnitude, can often be attributed to hyperparameter search schemes, random seeds, the number of models compared, etc. (Yogatama and Smith, 2015;Dodge et al., 2019). Thus, while model comparisons are an acceptable starting point for demonstrating whether or not a model is learning an interesting set of (or any!) cross-modal factors, they provide rather indirect evidence.
We propose Empirical Multimodally-Additive 1 function Projection (EMAP) as an additional diagnostic for analyzing multimodal classification models. Instead of comparing two different models, a single multimodal classifier's predictions are projected onto a less-expressive space: the result is equivalent to a set of predictions made by the closest possible ensemble of text-only and visualonly classifiers. The projection process is computationally efficient, has no hyperparameters to tune, can be implemented in a few lines of code, is provably unique and optimal, and works on any multimodal classifier: we apply it to models ranging from polynomial kernel SVMs to deep, pretrained, Transformer-based self-attention models.
We first verify that EMAPs do degrade performance for synthetic cases and for visual question answering cases where datasets have been specifically designed to require cross-modal reasoning. But we then examine a test suite of several recently-proposed multimodal prediction tasks that have not been specifically balanced in this way. We first achieve state-of-the-art performance for all of the datasets using a linear model. Next, we examine more expressive interactive models, e.g., pretrained Transformers, capable of cross-modal attention. While these models sometimes outperform the linear baseline, EMAP reveals that performance gains are (usually) not due to multimodal interactions being leveraged. Takeaways: For future work on multimodal classification tasks, we recommend authors report the performance of: 1) unimodal baselines; 2) any multimodal models they consider; and, critically, 3) the empirical multimodally-additive projection (EMAP) of their best performing multimodal model (see §6 for our full recommendations). 1 In §3, we more formally introduce additivity.

Related Work
Constructed multimodal classification tasks. In addition to image question answering/reasoning datasets already mentioned in §1, other multimodal tasks have been constructed, e.g., video QA (Lei et al., 2018;Zellers et al., 2019), visual entailment (Xie et al., 2018, hateful multimodal meme detection (Kiela et al., 2020), and tasks related to visual dialog (de Vries et al., 2017). In these cases, unimodal baselines are shown to achieve lower performance relative to their expressive multimodal counterparts. Collected multimodal corpora. Recent computational work has examined diverse multimodal corpora collected from in-vivo social processes, e.g., visual/textual advertisements (Hussain et al., 2017;Ye and Kovashka, 2018;Zhang et al., 2018), images with non-literal captions in news articles (Weiland et al., 2018), and image/text instructions in cooking how-to documents (Alikhani et al., 2019). In these cases, multimodal classification tasks are often proposed over these corpora as a means of testing different theories from semiotics (Barthes, 1988;O'Toole, 1994;Lemke, 1998;O'Halloran, 2004, inter alia); unlike many VQA-style datasets, they are generally not specifically balanced to force models to learn crossmodal interactions.
Without rebalancing, should we expect crossmodal interactions to be useful for these multimodal communication corpora? Some semioticians posit: yes! Meaning multiplication (Barthes, 1988) between images and text suggests, as summarized by Bateman (2014): under the right conditions, the value of a combination of different modes of meaning can be worth more than the information (whatever that might be) that we get from the modes when used alone. In other words, text 'multiplied by' images is more than text simply occurring with or alongside images. Jones et al. (1979) provide experimental evidence of conditional, compositional interactions between image and text in a humor setting, concluding that "it is the dynamic interplay between picture and caption that describes the multiplicative relationship" between modalities. Taxonomies of the specific types of compositional relationships image-text pairs can exhibit have been proposed (Marsh and Domas White, 2003;Martinec and Salway, 2005).
Model Interpretability. In contrast to methods that design more interpretable algorithms for prediction (Lakkaraju et al., 2016;Ustun and Rudin, 2016), several researchers aim to explain the behavior of complex, black-box predictors on individual instances (Štrumbelj and Kononenko, 2010;Ribeiro et al., 2016). The most related of these methods to the present work is Ribeiro et al. (2018), who search for small sets of "anchor" features that, when fixed, largely determine a model's output prediction on the input points. While similar in spirit to ours and other methods that "control for" a fixed subset of features (e.g., Breiman (2001)), their work 1) focuses only on high-precision, local explanations on single instances; 2) doesn't consider multimodal models (wherein feature interactions are combinatorially more challenging in comparison to unimodal models); and 3) wouldn't guarantee consideration of multimodal "anchors."

EMAP
Background. We consider models f that assign scores to textual-visual pairs (t, v), where t is a piece of text (e.g., a sentence), and v is an image. 2 In multi-class classification settings, values f (t, v) ∈ R d typically serve as per-class scores.
In ranking settings f (t 1 , v 1 ) may be compared to f (t 2 , v 2 ) via a ranking objective.
We are particularly interested in the types of compositionality that f uses over its visual and textual inputs to produce scores. Specifically, we distinguish between additive models (Hastie and Tibshirani, 1987) and interactive models (Friedman, 2001;Friedman and Popescu, 2008). 3 A function f of a representation z of n input features is additive in I if it decomposes as where (setting I = {1, . . . , n} for convenience) I ⊂ I indexes a subset of the features, \I = I \ I, and for any S ⊂ I, z S is the restriction of z to only those features whose indices appear in S.
In our case, because features are the union of Textual and Visual predictors, we say that a model is multimodally additive if it decomposes as Additive multimodal models are simply ensembles of unimodal classifiers and as such, may be considered underwhelming to multiple communities. A recent ACL paper, for example, refers to such ensembles as "naive." On the semiotics side, the conditionality implied by meaning multiplication (Barthes, 1988) -that the joint semantics of an image/caption depends non-linearly on its accompaniment -cannot be modeled additively: multimodally additive models posit that each image, independent of its text pairing, contributes a fixed score to per-class logits (and vice versa).
In contrast, multimodally interactive models are the set of functions that cannot be decomposed as in Equation 1 -that is, f 's output conditionally depends on its inputs in a non-additive fashion. Machine learning models. One canonical multimodally additive model is a linear model trained over a concatenation of textual and visual features . (2) We later detail several multimodally interactive models, including multi-layer neural networks, polynomial kernel SVMs, pretrained Transformer attention-based models, etc. However, even though interactive models are theoretically capable of modeling a more expressive set of relationships, it's not clear that they will learn to exploit this additional expressivity, particularly when simpler patterns suffice. Empirical multimodally-additive projections: EMAP. Given a fixed, trained model f (usually one theoretically capable of modeling visualtextual interactions) and a set of N labelled data- we aim to answer: does f utilize cross-modal feature interactions to produce more accurate predictions for these datapoints?
Hooker (2004) gives a general method for computing the projection of a function f onto a set of functions with a specified ANOVA structure (see also Sobol (2001); Liu and Owen (2006)): our algorithmic contributions are to extend the method to multimodal models with d > 1 dimensional outputs, and to prove that the multimodal empirical approximation is optimal. Specifically: we are Algorithm 1 Empirical Multimodally-Additive Projection (EMAP); worked example in Appendix G.
Input: a trained model f that outputs logits; a set of text/visual pairs Output: the predictions off , the empirical projection of f onto the set of multimodally-additive functions, on D.
for return preds f interested inf , the following projection of f onto the set of multimodally-additive functions: 4 and similarly for the other expectations. An empirical approximation of the partial dependence function for a given t i can be computed by looping over all observations: We similarly arrive atf V (v i ) andμ, yieldinĝ which is what EMAP, Algorithm 1, computes for each (t i , v i ). Note that Algorithm 1 involves evaluating the original model f on all N 2 v i , t j pairs -even mismatched image/text pairs that do not occur in the observed data. In practice, we recommend only computing this projection over the evaluation set. 5 Once the predictions of f andf are computed over the evaluation points, then their performance can be compared according to standard evaluation metrics, e.g., accuracy or AUC.
In Appendix A, we prove that the (meancentered) sum of empirical partial dependence functions is optimal with respect to squared error, that is: Claim. Subject to the constraint thatf is multimodally additive, Algorithm 1 produces a unique and optimal solution to arg min 4 Sanity Check: EMAP Hurts in Tasks that Require Interaction In §5.1, we will see that EMAP provides a very strong baseline for "unbalanced" multimodal classification tasks. But first, we first seek to verify that EMAP degrades model performance in cases that are designed to require cross-modal interactions.

Return the data point (V v, T t, y).
This function challenges models to learn whether or not the dot product of two randomly sampled vectors in d dimensions is positive or negative -a task that, by construction, requires modeling the multiplicative interaction of the features in these vectors. To complicate the task, the vectors are randomly projected to two "modalities" of different dimensions, d 1 and d 2 respectively.
We trained a linear model, a polynomial kernel SVM, and a feed-forward neural network on this data: the results are in Table 1. The linear model is additive and thus incapable of learning any meaningful pattern on this data. In contrast, the kernel SVM and feed-forward NN, interactive models, are able to fit the test data almost perfectly. However, when we apply EMAP to the interactive models, as expected, their performance drops to random.

Balanced VQA Tasks
Our next sanity check is to verify that EMAP hurts the performance of interactive models on two real multimodal classification tasks that are specifically balanced to require modeling cross-modal feature interactions: VQA 2.0 (Goyal et al., 2017) and GQA (Hudson and Manning, 2019). First, we fine-tuned LXMERT (Tan and Bansal, 2019), a multimodally-interactive, pretrained, 14layer Transformer model (See §5 for full description) that achieves SOTA on both datasets. The LXMERT authors frame question-answering as a multi-class image/text-pair classification problem -3.1K candidate answers for VQA2, and 1.8K for GQA. In Table 2, we compare, in crossvalidation, the means of: 1) accuracy of LXMERT, 2) accuracy of the EMAP of LXMERT, and 3) accuracy of simply predicting the most common answer for all questions. As expected, EMAP decreases accuracy on VQA2/GQA by 30/19 absolute accuracy points, respectively: this suggests LXMERT is utilizing feature interactions to produce more accurate predictions on these datasets. On the other hand, performance of LXMERT's EMAP remains substantially better than constant prediction, suggesting that LXMERT's logits do nonetheless leverage some unimodal signals in this data. 7

"Unbalanced" Datasets + Tasks
We now return to our original setting: multimodal classification tasks that have not been specifically formulated to force cross-modal interactions. Our goal is to explore what additional insights EMAP can add on top of standard model comparisons.
We consider a suite of 7 tasks, summarized in Table 3. These tasks span a wide variety of 7 EMAPed LXMERT's performance is comparable to LSTM-based, text-only models, which achieve 44.3/41.1 accuracy on the full VQA2/GQA test set, respectively.  Table 2: As expected, for VQA2 and GQA, the mean accuracy of LXMERT is substantially higher than its empirical multimodally additive projection (EMAP). Shown are averages over k = 15 random subsamples of 500 dev-set instances.
goals, sizes, and structures: some aim to classify semiotic properties of image+text posts, e.g., examining the extent of literal image/text overlap (I-SEM, I-CTX, T-VIS); others are post-hoc annotated according to taxonomies of social interest (I-INT, T-ST1, T-ST2); and one aims to directly predict community response to content (R-POP). 8 In some cases, the original authors emphasize the potentially complex interplay between image and text: Hessel et al. (2017) wonder if "visual and the linguistic interact, sometimes reinforcing and sometimes counteracting each other's individual influence;" Kruk et al.
(2019) discuss meaning multiplication, emphasizing that "the text+image integration requires inference that creates a new meaning;" and Vempala and Preoţiuc-Pietro (2019) attribute performance differences between their "naive" additive baseline and their interactive neural model to the notion that "both types of information and their interaction are important to this task." Our goal is not to downplay the importance of these datasets and tasks; it may well be the case that conditional, compositional interactions occur between images and text, but the models we consider do not yet take full advantage of them (we return to this point in item 6). Our goal, rather, is to provide diagnostic tools that can provide additional clarity on the remaining shortcomings of current models and reporting schemes.

Additive and Unimodal Models
The additive model we consider is the linear model from Equation 2 trained over the concatenation of image and text representations. 9 To represent im-  Fine-tuning a Pretrained Transformer. We fine-tuned LXMERT (Tan and Bansal, 2019) for our tasks. LXMERT represents images using 36 predicted bounding boxes, each of which is asregularization strength (10**(-7,-6,-5,-4,-3,-2,-1,0,1.0)), and loss type (logistic vs. squared hinge). We train models using lightning (Blondel and Pedregosa, 2016). 10 For reproducibility, we used ResNet-18 features for the Kruk et al. (2019) datasets; more detail in Appendix E. 11 Feature extraction approaches are known to produce competitive results relative to full fine-tuning (Devlin et al., 2019, §5.3); in some cases, mean pooling has been found to be similarly competitive relative to LSTM pooling (Hessel and Lee, 2019). 12 We again use grid search to optimize: polynomial kernel degree (2 vs. 3), regularization strength (10**(-5,-4,-3,-2,-1,0)), and gamma (1, 10, 100). 13 Parameters are optimized with the Adam optimizer (Kingma and Ba, 2015). We decay the learning rate when validation loss plateaus. The hyperparameters optimized in grid search are: number of layers (2, 3, 4), initial learning rate (.01, .001, .0001), activation function (relu vs. gelu), hidden dimension (128, 256), and batch norm (use vs. don't). Text is processed by a 9-layer Transformer. Finally, a 5-layer, crossmodal transformer processes the outputs of these unimodal encoders jointly, allowing for feature interactions to be learned. LXMERT's parameters are pre-trained on several cross-modal reasoning tasks, e.g., masked image region/language token prediction, cross-modal matching, and visual question answering. LXMERT achieves high performance on balanced tasks like VQA 2.0: thus, we know this model can learn compositional, cross-modal interactions in some settings. 14 LXMERT + logits: We also trained versions of LXMERT where fixed logits from a pretrained linear model (described above) are fed in to the final classifier layer. In this case, LXMERT is only tasked with learning the residual between the strong additive model and the labels. Our intuition was that this augmentation might enable the model to focus more closely on learning interactive, rather than additive, structure in the finetuning process.

Results
Our main prediction results are summarized in Table 4. For all tasks, the performance of our baseline additive linear model is strong, but we are usually able to find an interactive model that outperforms this linear baseline, e.g., in the case of T-ST2, a polynomial kernel SVM outperforms the linear model by 4 accuracy points. This observation alone seems to provide evidence that models  We utilize EMAP as an additional model diagnostic by projecting the predictions of our bestperforming interactive models. Surprisingly, for I-INT, I-SEM, I-CTX, T-ST1, T-ST2, and R-POP, EMAP results in essentially no performance degradation. Thus, even (current) expressive interactive models are usually unable to leverage cross-modal feature interactions to improve performance. This observation would be obfuscated without the use of additive projections, even though we compared to a strong linear baseline that achieves state-of-the-art performance for each dataset. This emphasizes the importance of not only comparing to additive/linear baselines, but also to the EMAP of the best performing model.
In total, for these experiments, we observe a single case, LXMERT + Linear Logits trained on T-VIS, wherein modeling cross-modal feature interactions appears to result in noticeable performance increases -here, the EMAPed model is 2.4 F1 points worse.
How much does EMAP change a model's predictions? For these datasets, a model and its EMAP usually make very similar predictions. For all datasets except T-VIS (where EMAP degrades performance) the best model and its EMAP agree on the most likely label in more than 95% of cases on average. For comparison, retraining the full models with different random seeds results in a 96% agreement on average. (Full results are in Appendix C.) 6 Implication FAQs Q1: What is your recommended experimental workflow for future work examining multimodal classification tasks? A: With respect to automated classifiers, we recommend reporting the performance of:

A constant and/or random predictor
to provide perspective on the interplay between the label distribution and the evaluation metrics. 2. As-strong-as-possible single-modal models f (t, v) = g(t) and f (t, v) = h(v) to understand how well the task can be addressed unimodally with present techniques.
3. An as-strong-as-possible multimodally addi- to understand multimodal model performance without access to sophisticated cross-modal reasoning capacity. 4. An as-strong-as-possible multimodally interactive model, e.g., LXMERT, to push predictive performance as far as possible. 5. The EMAP of the strongest multimodally interactive model. to determine whether or not the best interactive model is truly using cross-modal interactions to improve predictive performance.
We Long A: A skeptical reader might argue that, while multimodally-additive models cannot account for cross-modal feature interactions, such feature interactions may not be required for crossmodal reasoning. While we authors are not aware of an agreed-upon definition, 15 we will assume that "cross-modal reasoning tasks" are those that challenge models to compute (potentially arbitrary) logical functions of multimodal inputs. Under this assumption, we show that additive models cannot fit (nor well-approximate) most non-trivial logical functions. Consider the case of multimodal boolean target functions f (t, v) ∈ {0, 1}, and assume our image/text input feature vectors each consist of n binary features, i.e., t, v = t 1 , . . . t n , v 1 , . . . v n where t i , v i ∈ {0, 1}. Our goal will be to measure how well multimodally additive models can fit arbitrary logical functions f . To simplify our analysis in this idealized case: we assume 1) access to a training set consisting of all 2 2n input vector pairs, and only measure training accuracy (vs. the harder task of generalization) and 2) "perfect" unimodal representations in the sense that t, v contain all of the information required to compute f (t, v) (this is not the case for, e.g., CNN/RoBERTa features for actual tasks).
For very small cross-modal reasoning tasks, additive models can suffice. At n = 1 , there are 16 possible binary functions f (t 1 , v 1 ), and 14/16 can be perfectly fit by a function of the form f T (t 1 ) + f V (v 1 ) (the exceptions being XOR and XNOR). For n = 2, non-trivial functions are still often multimodally additively representable; an arguably surprising example is this one: But for cross-modal reasoning tasks with more variables, 16 even in this toy setting, multimodally-15 Suhr et al. (2019), for example, do not define "visual reasoning with natural language," but do argue that some tasks offer a promising avenue to study "reasoning that requires composition" via visual-textual grounding. 16 As a lower bound on the number of variables required in a more realistic case, consider a cross-modal reasoning task where questions are posed regarding the counts of, say, 1000 object types in images. It's likely that a separate image variable representing the count of each possible type of object would be required. While an oversimplification, for most cross-modal reasoning tasks, 2n = 6 variables is still relatively small. additive models ultimately fall short: for the n = 3 case, we sampled over 5 million random logical circuits and none were perfectly fit by the additive models. To better understand how well additive models can approximate various logical functions, we fit two types of them for the n > 2 case: 1) the EMAP of the input function f directly, and 2) AdaBoost (Freund and Schapire, 1995) with the weak learners restricted to unimodal models. 17 For a reference interactive model, we employ AdaBoost without the additivity restriction.  Figure 2 plots the AUC of the 3 models on the training set; we sample 10K random logical circuits for each problem size.
As the number of variables increases, the performance of the additive models quickly decreases (though full Ad-aBoost gets 100% accuracy in all cases). While these experiments are only for a toy case, they show that EMAPs, and additive models generally, have very limited capacity to compute or approximate logical functions of multimodal variables.

Conclusion and Future Work
The last question on our FAQ list in §6 leaves us with the following conundrum: 1) Additive models are incapable of most cross-modal reasoning; but 2) for most of the unbalanced tasks we consider, EMAP finds an additive approximation that makes nearly identical predictions to the full, interactive model. We postulate the following potential explanations, pointing towards future work: • Hypothesis 1: These unbalanced tasks don't require complex cross-modal reasoning. This purported conclusion cannot account for gaps between human and machine performance: if an additive model underperforms relative to human judgment, the gap could plausibly be explained by cross-modal feature interactions. But even in cases where an additive model matches or exceeds human performance on a fixed dataset, additive models may still be insufficient. The mere fact that unimodal and additive models can often 17 AdaBoost is chosen because it has strong theoretical guarantees to fit to training data: see Appendix D. be disarmed by adding valid (but carefully selected) instances post hoc (as in, e.g., Kiela et al. (2020)) suggests that their inductive bias can simultaneously be sufficient for train/test generalization, but also fail to capture the spirit of the task. Future work would be well-suited to explore 1) methods for better understanding which datasets (and individual instances) can be rebalanced and which cannot; and 2) the non-trivial task of estimating additive human baselines to compare against. • Hypothesis 2: Modeling feature interactions can be data-hungry. Jayakumar et al. (2020)  Concluding thoughts. Our hope is that future work on multimodal classification tasks report not only the predictive performance of their best model + baselines, but also the EMAP of that model. EMAP (and related algorithms) has practical implications beyond image+text classification: there are straightforward extensions to nonvisual/non-textual modalities, to classifiers using more than 2 modalities as input, and to singlemodal cases where one wants to check for feature interactions between two groups of features, e.g., premise/hypothesis in NLI.
Focused Research Award. Any opinions, findings, conclusions, or recommendations expressed here are those of the authors and do not necessarily reflect the view of the sponsors.

A Theoretical Analysis of Algorithm 1
We assume we are given a (usually evaluation) and a trained model f that maps t i , v i pairs to a d-dimensional vector of scores. We seek a multimodally-additive functionf that matches the values of f on any t i , v j for which there exist v , t such that t i , v ∈ D and t , v j ∈ D; that is, t i , v j represents any text-image pair we could construct if we decoupled the existing pairs in D. 18 For simplicity, first assume that d = 1 (we handle the d > 1 case later). Sincef is multimodally-additive by assumption, Proof. The first-order partial derivatives are: and the Hessian H is It suffices to show that H is positive semi-definite, i.e., for any z ∈ R 2n , z T Hz ≥ 0. Indeed, 18 Note that multimodally-additive models do not rely on particular ti, vj couplings, as this family of functions decomposes as f (ti, vj) = ft(ti) + fv(vj); thus, a multimodallyadditivef should be able to fit any image-text pair we could construct from D, not just the image-text pairs we observe.
Now, for the optimal solutions to our minimization problem, we can set the first-order partial derivatives to 0 and solve for our 2n parameters τ i , φ j . These solutions will correspond to global minima due to the convexity result we established above. It turns out to be equivalent to find solutions to: We can do this by finding one solution s to the above, and then analyzing the nullspace of H, which will turn out to be the 1-dimensional subspace spanned by r = 1, 1, . . . 1, n −1, −1, . . . − 1 n That Hr = 0 can be verified by direct calculation. Then, all solutions will have the form s + cr for any c ∈ R. 19 Claim 2. Algorithm 1 computes a solution to Equation 6 as a byproduct.
Proof. Algorithm 1 computes s = τ i , φ j as: s is a solution to Equation 6, as can be verified by direct substitution.
Claim 3. The rank of H is 2n − 1, which implies that its nullspace is 1-dimensional.
Proof. Solutions to Hx = λx occur when: So, zero is an eigenvalue of H with multiplicity 1, which shows that H's rank is 2n − 1. 20 Because the nullspace of H is 1-dimensional, all solutions to Equation 6 are given by s + cr for any c ∈ R. Returning to the notation of the original problem, we see that all optimal solutions are given by: 19 Proof: Assume that s is a solution of Equation 6. s − s will be in the nullspace of H. Clearly, s = s + (s − s), so s can be written as s + x for x in the nullspace of H. 20 We can make explicit the eigenbasis for the λ = n solutions. Let M ∈ R n×(n−1) be In−1 with an additional row of −1 concatenated as the n th row. The eigenbasis is given by the columns of M 0 0 M . Claim 4. Algorithm 1 produces a unique solution for the values off .
Proof. We have shown that Algorithm 1 produces an optimal solution, and have derived the parametric form of all optimal solutions in Equation 9 and Equation 10. Note that Algorithm 1 outputs τ i +φ j (rather than τ i , φ j individually). This cancels out the free choice of c. Thus, any algorithm that outputs optimal τ i + φ j will have the same output as Algorithm 1.
Extension to multiple dimensions. So far, we have shown that Algorithm 1 produces an optimal and unique solution for Equation 4, but only in cases where f ij , τ i , φ j ∈ R. In general, we need to show the algorithm works for multi-dimensional outputs, too. The full loss function includes a summation over dimension as: This objective decouples entirely over dimension d, i.e., the loss can be rewritten as: Furthermore, notice that the parameters in L i are disjoint from the parameters in L j if i = j. Thus, to minimize the multidimensional objective in Equation 11, it suffices to minimize the objective for each L i independently, and recombine the solutions. This is precisely what Algorithm 1 does.

B Dataset Details and Reproducibility Efforts B.1 I-INT, I-SEM, I-CTX
This data is available from https://github. com/karansikka1/documentIntent_ emnlp19. We use the same 5 random splits provided by the authors for evaluation. The authors provide ResNet18 features, which we use for our non-LXMERT experiments instead of EfficientNet-B4 features. After contacting the authors, they extracted bottom-up-top-down FasterRCNN features for us, so we were able to compare to LXMERT. State of the art performance numbers are derived from the above github repo; these differ slightly from the values reported in the original paper because the github versions are computed without image data augmentation.

B.2 T-VIS
This data is available from https: //github.com/danielpreotiuc/textimage-relationship/. The raw images are not available, so we queried the Twitter API for them. The corpus has 4472 tweets in it initially, but we were only able to re-collect 3905 tweets (87%) when we re-queried the API. Tweets can be missing for a variety of reasons, e.g., the tweet being permanently deleted, or the account's owner making the their account private at the time of the API request. A handful of tweets were available, but were missing images when we tried to re-collect them. This can happen when the image is a link to an external page, and the image is deleted from the external page.

B.3 R-POP
This data is available from http: //www.cs.cornell.edu/˜jhessel/ cats/cats.html. We just use the pics subreddit data. We attempted to rescrape the pics images from the imgur urls. We were able to re-collect 87215/88686 of the images (98%). Images can be missing if they have been, e.g., deleted from imgur. We removed any pairs with missing images from the ranking task; we trained on 42864/44343 (97%) of the original pairs. The data is distributed with training/test splits. From the training set for each split, we reserve 3K pairs for validation. The state of the art performance numbers are taken from the original releasing work.

B.4 T-ST1
This data is available from http: //www.ee.columbia.edu/ln/dvmm/ vso/download/twitter_dataset.html and consists of 603 tweets (470 positive, 133 negative). The authors distribute data with 5 folds pre-specified for cross-validation performance reporting. However, we note that the original paper's best model achieves 72% accuracy in this setting, but a constant prediction baseline achieves higher performance: 470/(470+133) = 78%. Note that the constant prediction baseline likely performs worse according to metrics other than accuracy, but only accuracy is reported. We attempted to contact the authors of this study but did not receive a reply. We also searched for additional baselines for this dataset, but were unable to find additional work that uses this dataset in the same fashion. Thus, given the small size of the dataset, lack of reliable measure of SOTA performance, and label imbalance, we decided to report ROC AUC prediction performance.

B.5 T-ST2
This data is available from https: //www.mcrlab.net/research/mvsasentiment-analysis-on-multiview-social-data/. We use the MVSA-Single dataset because human annotators examine both the text and image simultaneously; we chose not to use MVSA-Multiple because human annotators do not see the tweet's image and text at the same time. However, the dataset download link only comes with 4870 labels, instead of the 5129 described in the original paper. We contacted the authors of the original work about the missing data, but did not receive a reply. We follow the preprocessing steps detailed in Xu and Mao (2017) to derive a training dataset. After preprocessing, we are left with 4041 data points, whereas prior work compares with 4511 points after preprocessing. The preprocessing consists of removing points that are (positive, negative), (negative, positive), or (neutral, neutral), which we believe matches the description of the preprocessing in that work. We contacted the authors for details, but did not receive a reply. The state-of-the-art performance number for this dataset is from Xu et al. (2018).

C Are EMAPs just regularizers?
One reason why EMAPs may often offer strong performance is by acting as a regularizer: projecting to a less expressive hypothesis space may reduce variance/overfitting. But in many cases, the original model and its EMAP achieve similar predictive accuracy. This suggests two explanations: either the EMAPed version makes  significantly different predictions with respect the original model (e.g., because it is better regularized), but it happens that those differing predictions "cancel out" in terms of final prediction accuracy; or, the original, unprojected functions are quite close to additive, anyway, and the EMAP doesn't change the predictions all that much.
We differentiate between these two hypotheses by measuring the percent of instances for which the EMAP makes a different classification prediction than the full model. Table 5 gives the results: in all cases except T-VIS, the match between EMAP and the original model is above 95%. For reference, we retrained the best performing models with different random seeds, and measured the performance difference under this change.
When EMAP changes the predictions of the original model, does the projection generally change to a more accurate label, or a less accurate one? We isolate the instances where a label swap occurs, and quantify this using: (num orig better) / (num orig better + num proj better). In most cases, the effect of projecting improves and degrades performance roughly equally, at least according to this metric. For T-VIS, however, the original model is better in over 55% of cases: this is also reflected in the corresponding F-scores.

D Logic Experiment Details
In §6, we describe experiments using AdaBoost. We chose AdaBoost (Freund and Schapire, 1995) to fit to the training set because of its strong convergence guarantees. In short: if AdaBoost can find a weak learner at each iteration (that is: if it can find a candidate classifier with above-random performance) it will be able to fit the training set. A more formal statement of AdaBoost's properties can be found in the original work.
The AdaBoost classifiers we consider use decision trees with max depth of 15 as base estimators. We choose a relatively large depth because we are not concerned with overfitting: we are just measuring training fit. The additive version of AdaBoost we consider is identical to the full AdaBoost classifier, except, at each iteration, either the image features or the text features individually are considered.

E Additional Reproducibility Info
Computing Hardware.
Linear models and feed-forward neural networks were trained using consumer-level CPU/RAM configurations. LXMERT was fine-tuned on single, consumer GPUs with 12GB of vRAM. Runtime. The slowest algorithm we used was LXMERT (Tan and Bansal, 2019), and the biggest  Figure 3: Examples of cases from for which EMAP degrades the performance of LXMERT + Logits. All cases are labelled as "image does not add meaning" in the original corpus. Text of tweets may be gently modified for privacy reasons.
dataset we fine-tuned on was R-POP. LXMERT was fine-tuned on the order of 1500 times. Depending on the dataset, the fine-tuning process took between 10 minutes and an hour. Overall, we estimate that we spent on the order of 50-100 GPU days doing this work. Number of Parameters. The RoBERTa features we used were 2048-dimensional, and the Efficient-NetB4 features were 1792 dimensional. The linear models and feed forward neural networks operated on those. We cross-validated the number of layers and the width, but the maximal model, for these cases, has on the order of millions of parameters. The biggest model we used was LXMERT, which has a comparable memory footprint to the original BERT Base model.

F Qualitative Analysis of T-VIS
To demonstrate the potential utility of EMAP in qualitative examinations, we identified the individual instances in T-VIS for which EMAP changes the test-time predictions of the LXMERT + Linear Logits model. Recall that in this dataset, EMAP hurts performance.
In introducing this task, Vempala and Preoţiuc-Pietro (2019) propose categorizing image+text tweets into four categories: "Some or all of the content words in the text are represented in the image" (or not) × "Image has additional content that represents the meaning of the text and the image" (or not).
As highlighted in Table 5, when EMAP changes the prediction of the full model (14.5% of cases), the prediction made is incorrect more often not: among label swapping cases, when the EMAP or the original model is correct, the original prediction is correct in 55% of the cases.
The most common label swaps of this form are between the classes: "image does not add" × {"text is represented", "text is not represented"}; as shorthand for these two classes, we will write IDTR ("image doesn't add, text represented") and IDTN ("image doesn't add, text not represented"  Figure 3 gives some instances from T-VIS where images do not add meaning and the EMAP projection causes a swap from a correct to an incorrect prediction. While it's difficult to draw conclusions from single instances, some preliminary patterns emerge. There are a cluster of animal images coupled with captions that may be somewhat difficult to map. In the case of the dog with eyebrows, the image isn't centered on the animal, and it might be difficult to identify the eyebrows without the prompt of the caption (hence, interactions might be needed). Similarly, the bear-lookingdog case is difficult: the caption doesn't explicitly mention a dog, and the image itself depicts an animal that isn't canonically canine-esque; thus, modeling interactions between the image and the caption might be required to fully disambiguate the meaning. Figure 3 also depicts two cases where the original model predicts that the text is not represented (but EMAP does). They are drawn from a cluster of similar cases where the captions seem indirectly connected to the image. Consider the 4 th , music composition example: just looking at the text, one could envision a more literal manifestation: i.e., a person playing a horn. Similarly, looking just at the screenshot of the music production software, one could envision a more literal caption,