v^{\mathfrak{k}},\iota(p).f_{-}\right\rangle$ $\displaystyle=\left\langle
v^{\mathfrak{k}},\iota(p).f_{-}\right\rangle.$

Let us write $p=\sum\prod a_{i}^{n_{i}}$ as a polynomial in
$\mathfrak{S(a)}\cong\mathfrak{P}(\mathfrak{a}^{*})$ (see (27)). Then
$\iota(p)=\sum\prod\iota\left(a_{i}^{n_{i}}\right)=\sum\prod(-1)^{n_{i}}a_{i}^{n_{i}}$
as all the terms commute. This gives

$\displaystyle\left\langle v^{\mathfrak{k}},\iota(p).f_{-}\right\rangle$
$\displaystyle=\sum\prod\left\langle
v^{\mathfrak{k}},(-1)^{n_{i}}a_{i}^{n_{i}}.f_{-}\right\rangle$
$\displaystyle=\sum\prod(-1)^{n_{i}}\left(a_{i}\left(-\lambda\right)\right)^{n_{i}}\left\langle
v^{\mathfrak{k}},f_{-}\right\rangle$
$\displaystyle=\sum\prod(-1)^{n_{i}}(-1)^{n_{i}}a_{i}\left(\lambda\right)\left\langle
v^{\mathfrak{k}},f_{-}\right\rangle$
$\displaystyle=p\left(\lambda\right)\left\langle
v^{\mathfrak{k}},f_{-}\right\rangle.$

Since $\left\langle v^{\mathfrak{k}},f_{-}\right\rangle\neq 0$ by Proposition
6.4, the calculation yields
$C=p\left(\lambda\right)=\pi(D)\left(\lambda\right)$. Finally, Definition 2.4
gives $C=\Gamma(D)\left(\lambda+\rho\right)$. ∎

###  6.2. The Vanishing Condition

In this subsection, we prove the main result with some weight module theory
machinery. Throughout the subsection, we assume Conjecture 1. That is,
$V\left(\lambda^{\natural}\right)$ is spherical, whose highest weight on
$\mathfrak{h}$ is $2\lambda_{\mathfrak{h}}^{\natural}$ (33).

First, we recall the following “weight decomposition” proposition. This is
parallel to Proposition 3.2 in [Kum10]. In the following proposition, we let
$\mathfrak{L}$ be a Lie (super)algebra, $\mathfrak{b}\subseteq\mathfrak{L}$ be
a Borel subalgebra. In a non-super setting, one could prove it using
Steinberg’s formula, c.f. [Hum78].

######  Proposition 6.6.

Let $W(\mu),W(\lambda)$ be two finite dimensional irreducible
$\mathfrak{L}$-modules with highest weight $\mu$ and $\lambda$, and $V$ be a
weight module. If
$\operatorname{Hom}_{\mathfrak{L}}\left(W\left(\mu\right),W\left(\lambda\right)\otimes
V\right)\neq\\{0\\}$, then $\mu=\lambda+\nu$ for some weight $\nu\in\Pi(V)$.

###### Proof.

Denote the one-dimensional $\mathfrak{b}$-module of weight $\varepsilon$ as
$\mathbb{C}_{\varepsilon}$. Then we have

$\displaystyle\operatorname{Hom}_{\mathfrak{L}}\left(W\left(\mu\right),W\left(\lambda\right)\otimes
V\right)$
$\displaystyle\cong\operatorname{Hom}_{\mathfrak{b}}\left(\mathbb{C}_{\mu},W\left(\lambda\right)\otimes
V\right)$
$\displaystyle\cong\operatorname{Hom}_{\mathfrak{b}}\left(\mathbb{C}_{\mu}\otimes
W^{*}\left(\lambda\right),V\right).$

Notice that $W^{*}\left(\lambda\right)$, as a $\mathfrak{b}$-module, is
generated by some lowest vector $w_{-\lambda}\in\mathbb{C}_{-\lambda}$. Hence
any non-zero $\mathfrak{b}$-invariant map $\phi:\mathbb{C}_{\mu}\otimes
W^{*}\left(\lambda\right)\rightarrow V$ is completely determined by its
restriction on $\mathbb{C}_{\mu}\otimes\mathbb{C}_{-\lambda}$ which is of
weight $\mu-\lambda$. But any such $\phi$ must preserve weights. Therefore
$\mu-\lambda\in\Pi(V)$. ∎

Now we can prove a branching lemma in the spirit of Lemma 5.1 in [SZ19].

######  Lemma 6.7.

Let $\mu,\lambda\in\mathscr{H}(p,q)$. If $\mu\nsubseteq\lambda$, then
$\operatorname{Hom}_{\mathfrak{k}}\left(W\left(\mu^{\natural}\right),V\left(\lambda^{\natural}\right)\right)=0$.

###### Proof.

We show the contrapositive. Suppose $W\left(\mu^{\natural}\right)$ occurs as a
homomorphic image in $V\left(\lambda^{\natural}\right)$. Consider the induced
module
$I(\lambda):=\operatorname{Ind}_{\mathfrak{k}+\mathfrak{p}^{+}}^{\mathfrak{g}}W\left(\lambda^{\natural}\right)$.
Recall that the Borel subalgebra of $\mathfrak{g}$ is extended from the one of
$\mathfrak{k}$ by $\mathfrak{p}^{+}$, so $V\left(\lambda^{\natural}\right)$ is
the irreducible quotient of $I(\lambda)$. Therefore
$W\left(\mu^{\natural}\right)$ occurs in $I(\lambda)$ by assumption.

Now $I(\lambda)$ is also isomorphic to $\mathfrak{S}(\mathfrak{p}^{-})\otimes
W\left(\lambda^{\natural}\right)$ by the Poincaré–Birkhoff–Witt theorem. By
Proposition 6.6, if $W\left(\mu^{\natural}\right)$ occurs in
$\mathfrak{S}(\mathfrak{p}^{-})\otimes W\left(\lambda^{\natural}\right)\cong
W\left(\lambda^{\natural}\right)\otimes\mathfrak{S}(\mathfrak{p}^{-})$, then
$\mu^{\natural}=\lambda^{\natural}+\nu$ for some $\nu$ in
$\Pi(\mathfrak{S}(\mathfrak{p}^{-}))$. Here
$\nu=\mu^{\natural}-\lambda^{\natural}=\sum\nu_{i}\gamma^{\textsc{B/F}}_{i}$
for $\gamma_{i}^{\textsc{B}}=\epsilon_{i}^{+}-\epsilon_{i}^{-}$ and
$\gamma_{i}^{\textsc{F}}=\delta_{i}^{+}-\delta_{i}^{-}$. But the weights in
$\mathfrak{p}^{-}$ are in the form of
$\epsilon_{i}^{-}-\epsilon_{j}^{+},\delta_{i}^{-}-\delta_{j}^{+},\epsilon_{i}^{-}-\delta_{j}^{+},\delta_{i}^{-}-\epsilon_{j}^{+}$,
and the weights in $\mathfrak{S}(\mathfrak{p}^{-})$ are non-negative integral
combinations of them. In particular, the coefficients of $\epsilon_{i}^{+}$
and $\delta_{i}^{+}$ are always non-positive. Necessarily, $\nu_{i}\leq 0$
which is equivalent to $\mu\subseteq\lambda$. ∎

Recall $D_{\mu}$ is defined as follows:

$\displaystyle\operatorname{End}_{\mathfrak{k}}$
$\displaystyle\left(W\left(\mu^{\natural}\right)\right)\cong
W^{*}\left(\mu^{\natural}\right)\otimes
W\left(\mu^{\natural}\right)\hookrightarrow\mathfrak{U}\left(\mathfrak{p}^{-}\right)\otimes\mathfrak{U}\left(\mathfrak{p}^{+}\right)\rightarrow\mathfrak{U}\left(\mathfrak{g}\right)^{\mathfrak{k}}$
$\displaystyle
1\xmapsto{\hphantom{\operatorname{End}_{\mathfrak{k}}\left(W\left(\mu^{\natural}\right)\right)\cong
W^{*}\left(\mu^{\natural}\right)\otimes
W\left(\mu^{\natural}\right)\hookrightarrow\mathfrak{U}\left(\mathfrak{p}^{-}\right)\otimes\mathfrak{U}\left(\mathfrak{p}^{+}\right)\rightarrow\mathfrak{U}\left(\mathfrak{g}\right)^{\mathfrak{k}}\mathfrak{U}\left(\mathfrak{g}\right)^{\mathfrak{k}}}}D_{\mu}$

Let us take a basis for $W\left(\mu^{\natural}\right)$ consisting of
$\mathfrak{t}$-weight vectors

$\left\\{\xi_{\mu}^{\nu,i}:[T,\xi_{\mu}^{\nu,i}]=\nu(T)\xi_{\mu}^{\nu,i},\text{
for any }T\in\mathfrak{t}\right\\}_{\nu\leq\mu^{\natural}},$

and the dual basis
$\left\\{\eta_{\mu}^{-\nu,i}\right\\}_{\nu\leq\mu^{\natural}}$, then the super
Shimura operator associated with $\mu$ can be written as

(39) $D_{\mu}=\sum\eta_{\mu}^{-\nu,i}\xi_{\mu}^{\nu,i}.$

######  Theorem 6.8.

For $\mu\nsubseteq\lambda$,
$\Gamma(D_{\mu})\left(2\lambda_{\mathfrak{h}}^{\natural}+\rho\right)=0$.

###### Proof.

Let $f_{-}\in V^{*}\left(\lambda^{\natural}\right)$ be a non-zero lowest
weight functional and take the spherical vector $v^{\mathfrak{k}}$ in
$V\left(\lambda^{\natural}\right)$ as above. By Theorem 6.2, we know that the
pairing $\left\langle D_{\mu}.v^{\mathfrak{k}},f_{-}\right\rangle$ is equal to

$\Gamma(D_{\mu})\left(2\lambda_{\mathfrak{h}}^{\natural}+\rho\right)\left\langle
v^{\mathfrak{k}},f_{-}\right\rangle.$

On the other hand, by spelling out $D_{\mu}$, we have

(40) $\left\langle
D_{\mu}.v^{\mathfrak{k}},f_{-}\right\rangle=-\sum\left\langle\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}},\eta_{\mu}^{-\nu,i}.f_{-}\right\rangle.$

For each $\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}\in
V\left(\lambda^{\natural}\right)$, if we apply $X\in\mathfrak{k}$, we get

(41)
$X.\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}=[X,\xi_{\mu}^{\nu,i}].v^{\mathfrak{k}}\pm\xi_{\mu}^{\nu,i}.X.v^{\mathfrak{k}}=[X,\xi_{\mu}^{\nu,i}].v^{\mathfrak{k}}$

by the definition of $v^{\mathfrak{k}}$. Here we have $\pm$ due to different
parities.

Since $X\in\mathfrak{k}$ and $W\left(\mu^{\natural}\right)$ is a
$\mathfrak{k}$-module, we observe that $[X,\xi_{\mu}^{\nu,i}]$ stays in
$W\left(\mu^{\natural}\right)$, and the right side of (41) is therefore a
linear combination of various $\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}$. Therefore,
the space $S$ spanned by all $\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}$ is a
$\mathfrak{k}$-submodule of $V\left(\lambda^{\natural}\right)$. Also, if we
apply $T\in\mathfrak{t}$ on $\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}$, we get

$T.\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}=[T,\xi_{\mu}^{\nu,i}].v^{\mathfrak{k}}-\xi_{\mu}^{\nu,i}.T.v^{\mathfrak{k}}=\nu(T)\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}$

which means each $\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}$ is a weight vector in
$S$ for all the weights $\nu$ appearing in $W\left(\mu^{\natural}\right)$. By
(41), $S$ is a homomorphic image of $W\left(\mu^{\natural}\right)$ in
$V\left(\lambda^{\natural}\right)$. By Lemma 6.7, $S=\left\\{0\right\\}$, and
hence

$\xi_{\mu}^{\nu,i}.v^{\mathfrak{k}}=0$

for all $\xi_{\mu}^{\nu,i}$. Now the right-hand side of (40) is 0 and since
the pairing $\left\langle v^{\mathfrak{k}},f_{-}\right\rangle$ is non-zero by
Proposition 6.4, we have the desired result. ∎

###  6.3. Proof to Theorem 1.2

In Proposition 4.6, we see that the images of $D_{\mu}$ do land in
$\Lambda^{0}(\mathfrak{a}^{*})=\mathfrak{P}(\mathfrak{a}^{*})^{\mathfrak{W}}$.
This allows us to compare the (shifted) symmetric polynomials $I_{\mu}$ and
$\Gamma(D_{\mu})$, where we take $I_{\mu}$ in
$\Lambda_{\Sigma}^{-\frac{1}{2}\rho}$ as in Theorem 2.30.

######  Theorem 6.9 (Theorem 1.2).

The Harish-Chandra image of the super Shimura operator associated with $\mu$,
$\Gamma(D_{\mu})(X)\in\mathfrak{P}(\mathfrak{a}^{*})^{\mathfrak{W}}$, is equal
to a non-zero multiple of the $\tau$ image of the Type BC interpolation
supersymmetric polynomial $\tau(I_{\mu})\left(X;-1,p-q+\frac{1}{2}\right)$.

###### Proof.

By Proposition 4.6, the degree $2|\mu|$ polynomial $\tau(I_{\mu})$ is in
$\Lambda^{0}(\mathfrak{a}^{*})$ and satisfies the vanishing condition

$\tau(I_{\mu})\left(2\lambda_{\mathfrak{h}}^{\natural}+\rho;-1,p-q+\frac{1}{2}\right)=0$

for all $\lambda\nsupseteq\mu$.

As for $\Gamma(D_{\mu})$, it is clear that $\Gamma$ is degree-preserving and
the degree of $D_{\mu}$ is indeed $2|\mu|$ (39). Theorem 6.8 dictates the same
vanishing condition of $\tau(I_{\mu})$ for $\Gamma(D_{\mu})$. The uniqueness
of $I_{\mu}$, hence of $\tau(I_{\mu})$, shows that $\Gamma(D_{\mu})$ is
proportional to $\tau(I_{\mu})$.

To show the multiple is non-zero, it suffices to prove
$D_{\mu}\notin\mathfrak{k}\mathfrak{U}$, which is contained in the kernel of
$\Gamma$ by Theorem 2.5. We notice that $D_{\mu}$, by definition, is an
element in $\mathfrak{S}(\mathfrak{p}^{-})\mathfrak{S}(\mathfrak{p}^{+})$. By
the Poincaré–Birkhoff–Witt theorem, one may write

$\mathfrak{U}=\mathfrak{k}\mathfrak{U}\oplus\mathfrak{S}(\mathfrak{p}^{-})\mathfrak{S}(\mathfrak{p}^{+})$

and since $D_{\mu}$ is clearly non-zero, $\Gamma(D_{\mu})$ is also non-zero. ∎

######  Remark 6.10.

We explain why the condition for sphericity given in [AS15] (see Subsection
5.2) is insufficient for the above approach. Consider the case $p=q=1$. Direct
computations show $\rho=(-1,1)$. Let $\mu=(2)\in\mathscr{H}(1,1)$. The
vanishing condition of $\tau(I_{\mu})$ says it is zero for any $\lambda$ not
containing $(2)$, i.e. $\tau(I_{\mu}=0)$ for the partitions $(1^{n})$ for
$n\in\mathbb{N}$ (one-column partitions). By Proposition 5.4, the only
partition above guaranteed to give a spherical
$V\left(\lambda^{\natural}\right)$ is $\lambda=(1)$. So
$2\lambda_{\mathfrak{h}}^{\natural}+\rho=(1,1)$ which means
$\Gamma(D_{\mu})((1,1))=0$. Normalize $\Gamma(D_{\mu})\in P[x,y]$ so that the
resulting polynomial $f$ has leading coefficient 1. Then as a degree 4 even
supersymmetric polynomial in two variables, we may assume
$f(x,y)=(x^{2}-y^{2})(x^{2}+ay^{2}+b)$. We see that $f(1,1)=0$ for any $a$ and
$b$. In fact, for $\mu=(2)$, we have $f(x,y)=(x^{2}-y^{2})(x^{2}-1)$ which
indeed vanishes at $2\lambda_{\mathfrak{h}}^{\natural}+\rho=(1,2n-1)$ for all
$\lambda=(1^{n})\nsupseteq(2)$.

## Acknowledgement

The author would like to thank his advisor Siddhartha Sahi for his munificent
help in proposing the project and providing insights. The author would also
like to thank Shifra Reif for enlightening conversations. In the earlier stage
of the project, the author also received great help from Hadi Salmasian.

##  Appendix A Deformed Root System and Interpolation Polynomials

###  A.1. Admissible Deformed Root Systems

We introduce the deformed root systems studied by Sergeev and Veselov in
[SV04, SV09] in this subsection. These root systems are based on the
generalized root systems introduced by Serganova [Ser96].

Let $V=\mathbb{C}^{m|n}$ be a vector superspace with the standard basis
$\left\\{\epsilon_{1},\epsilon_{2},\dots,\epsilon_{m},\delta_{1},\dots,\delta_{n}\right\\}$
and $I_{\overline{0}}=\left\\{1,\dots,m\right\\}$ be the index set
corresponding to $\epsilon_{i}$’s and
$I_{\overline{1}}=\left\\{m+1,\dots,m+n\right\\}$ be the index set
corresponding to $\delta_{i}$’s. Write $I=I_{\overline{0}}\cup
I_{\overline{1}}$. We define a bilinear form on $V$ by

$B(u,v):=\sum_{i\in I_{\overline{0}}}u_{i}v_{i}-\sum_{j\in
I_{\overline{1}}}u_{j}v_{j}$

where $x_{i}$ denote the $i$-th coordinate of $x\in V$ so that
$x=\sum_{i=1}^{m}x_{i}\epsilon_{i}+\sum_{j=1}^{n}x_{m+j}\delta_{j}$.

We say a vector $v$ is isotropic (resp. anisotropic) if $B(v,v)=0$ (resp.
$\neq 0$). The two (generalized) root systems we consider in this paper are as
follows:

  1. (1)

$C(m,n)$: Let $C(I)=\left\\{\pm
2\epsilon_{i},\pm\epsilon_{i}\pm\epsilon_{j}:i,j\in I,i\neq j\right\\}$. We
define ${{}_{1}\Sigma}:=C(I_{\overline{0}})\cup C(I_{\overline{1}})$ for
anisotropic roots and
${{}_{0}\Sigma}=\left\\{\pm\epsilon_{i}\pm\delta_{j}:i\in
I_{\overline{0}},j\in I_{\overline{1}}\right\\}$ for isotropic roots. Then
$C(m,n)$ is defined as ${{}_{1}\Sigma}\cup{{}_{0}\Sigma}$.

  2. (2)

$BC(m,n)$: Let $BC(I)=\left\\{\pm\epsilon_{i},\pm
2\epsilon_{i},\pm\epsilon_{i}\pm\epsilon_{j}:i,j\in I,i\neq j\right\\}$. We
define ${{}_{1}\Sigma}:=BC(I_{\overline{0}})\cup BC(I_{\overline{1}})$ for
anisotropic roots and
$\Sigma_{\overline{1}}=\left\\{\pm\epsilon_{i}\pm\delta_{j}:i\in
I_{\overline{0}},j\in I_{\overline{1}}\right\\}$ for isotropic roots. Then
$BC(m,n)$ is defined as ${{}_{1}\Sigma}\cup{{}_{0}\Sigma}$.

When $n=0$, we get the usual type $C$ or type $BC$ systems. In [SV04], an
admissible deformation of these generalized root systems is introduced. This
appears in the study of the Calogero–Moser–Sutherland problem in which the
root system stays the same while the bilinear form and the multiplicities are
“tweaked”. The deformed bilinear form is

$B_{\mathsf{k}}(u,v):=\sum_{i\in
I_{\overline{0}}}u_{i}v_{i}+\mathsf{k}\sum_{i\in I_{\overline{1}}}u_{i}v_{i}.$

Five parameters regarding the multiplicities of the roots are presented as
follows:

$\operatorname{\texttt{mult}}\left(\pm\epsilon_{i}\pm\epsilon_{j}\right)=\mathsf{k},\operatorname{\texttt{mult}}\left(\pm\epsilon_{i}\right)=\mathsf{p},\operatorname{\texttt{mult}}\left(\pm
2\epsilon_{i}\right)=\mathsf{q};$
$\operatorname{\texttt{mult}}\left(\pm\delta_{i}\pm\delta_{j}\right)=\mathsf{k}^{-1},\operatorname{\texttt{mult}}\left(\pm\delta_{i}\right)=\mathsf{r},\operatorname{\texttt{mult}}\left(\pm
2\delta_{i}\right)=\mathsf{s};$

They satisfy the following two relations:

$\mathsf{p=kr},\mathsf{2q+1=k(2s+1)}.$

The multiplicities of isotropic roots ($\pm\epsilon_{i}\pm\delta_{j}$) are set
to be 1 (c.f. the parameter $q_{\overline{1}}$ in Remark 4.2), and the form
and all multiplicities stay $W_{0}$ invariant.

###  A.2. Interpolation Polynomials

We describe the discrepancy between our version of the Type $BC$ interpolation
polynomials and the ones that appear in Section 6, [SV09].

The following table of two different sets of notations is provided for the
convenience of our readers. Since we will alter the constructions and proofs
directly in [SV09], in what follows, we will use the notations given on the
left.

[SV09] |  Z.
---|---
$m,n$ |  $p,q$
$k,p,q,r,s$ |  $\mathsf{k,p,q,r,s}$
$m(-),\epsilon,\delta,R^{+},\rho$ |  $\operatorname{\texttt{mult}}(-),\alpha^{\textsc{B}},\alpha^{\textsc{F}},\Sigma^{+}_{\textup{SV}},\rho_{\textup{SV}}$
$w,z$ |  $x,y$

Unless otherwise indicated, all the numbered equations and results refer to
those in [SV09].

The origin of the discrepancy lies in a different choice of positivity of the
root system in [SV09]. Formula (71) reads
“$R^{+}_{iso}=\left\\{\delta_{p}\pm\epsilon_{i}\right\\}$”. That is, they
require $\alpha^{\textsc{F}}\pm\alpha^{\textsc{B}}$ in this paper to be
positive. We denote this positive system as $\Sigma^{+}_{\textup{SV}}$.

The half sum of positive roots “$\rho$” involved in their description is
defined as follows

$\rho_{\textup{SV}}:=\frac{1}{2}\sum_{\alpha\in\Sigma^{+}_{\textup{SV}}}m(\alpha)\alpha$

To prove Proposition 6.3, they use a special restriction map
$\mathrm{res}_{m,n}$ from $\Lambda^{\mathsf{k,h}}$, the usual ring of
symmetric functions of type $BC$, to $P_{m,n}=\mathbb{C}[w_{1},\dots,z_{n}]$.
The precise definition is as follows:

For any $\lambda\in\mathscr{H}(m,n)$, let $\nu$ be the transpose of the first
$n$ columns and $\mu$ the remaining part. Notice that this is different from
constructing $\lambda^{\natural}$ where we take the first $m$ rows and then
take the remaining part. Then we define a map
$\chi:\mathscr{H}(m,n)\rightarrow\mathbb{C}^{m+n}$ which sends $\lambda$ to
$(\mu_{1},\dots,\mu_{m},\nu^{\prime}_{1},\dots,\nu^{\prime}_{n})$. The image
of $\chi$ is Zariski dense and any symmetric function in
$\Lambda^{\mathsf{k,h}}$ can be restricted to a function on
$\mathscr{H}(m,n)$. Denote this restriction map as
$\mathrm{res}_{m,n}:\Lambda^{\mathsf{k,h}}\rightarrow P_{m,n}$. More
precisely, given $f\in\Lambda^{\mathsf{k,h}}$, $\mathrm{res}_{m,n}(f)$ is the
unique polynomial such that $f(\lambda)=\mathrm{res}_{m,n}(f)(\chi(\lambda))$
for any $\lambda\in\mathscr{H}(m,n)$.

The following theorem is shown using the even Bernoulli polynomials.

######  Theorem A.1.

For generic values of $\mathsf{k}$ (not equal to a positive rational number),
we have

$\operatorname{Im}\mathrm{res}_{m,n}=\Lambda_{m,n}^{\mathsf{k,h}}.$

Proposition 6.3 then follows from Theorem A.1 by applying $\mathrm{res}_{m,n}$
to the Okounkov interpolation polynomials in $\Lambda^{\mathsf{k,h}}$. That
is,

$\textup{Type $BC$ Supersymmetric Interpolation
Polynomial}=\mathrm{res}_{m,n}(\textup{Okounkov Polynomial})$

Both are denoted using $I_{\lambda}$.

The readers may recall that $\lambda^{\natural}$ is used in our paper
consistently rather than $\chi(\lambda)$. In fact, this subtlety is an
interesting combinatorial consequence of different choices of positivity. Let
us “fix” Theorem A.1 with our choice of positivity as follows:

  1. (1)

In (71), change $R^{+}_{iso}=\left\\{\delta_{p}\pm\epsilon_{i}\right\\}$ to
$R^{+}_{iso}=\left\\{\epsilon_{i}\pm\delta_{p}\right\\}$. This makes $\rho$
(72) equal to

$-\sum_{i=1}^{m}(h+ki)\epsilon_{i}-k^{-1}\sum_{j=1}^{n}\left(h+\frac{1}{2}k-\frac{1}{2}+j+km\right)\delta_{j}.$

  2. (2)

Consequently, the definition of the ring $\Lambda_{m,n}^{\mathsf{k,h}}$ is now
the subalgebra consisting of polynomials symmetric separately in
$(w_{i}+h+ki)^{2}$ and
$\left(z_{j}+hk^{-1}+\frac{1}{2}-\frac{1}{2}k^{-1}+jk^{-1}+m\right)^{2}$, and
satisfying the same supersymmetry condition, changing $\alpha$ to
$\epsilon_{i}-\delta_{p}$.

  3. (3)

For $\lambda$, bisect it by taking the first $m$ rows and the rest $n$
columns, so that we get $\lambda^{\natural}=:\chi(\lambda)$. Equivalently,
$w_{i}=\lambda_{i}$ and
$z_{j}=\left\langle\lambda_{j}^{\prime}-m\right\rangle$. Again, this new
version of $\chi$ has a Zariski dense image and $\mathrm{res}_{m,n}$ is well-
defined.

  4. (4)

The statement of Theorem 6.1 is correct verbatim, while in the proof, the
polynomial $f_{l}^{m,n}$ is now changed to

$\displaystyle f_{l}^{m,n}(w,z)=$
$\displaystyle\sum_{i=1}^{m}\left(B_{2l}\left(w_{i}+h+ki+\frac{1}{2}\right)-B_{2l}\left(h+ki+\frac{1}{2}\right)\right)$
$\displaystyle+k^{2i-1}\sum_{j=1}^{n}\left(B_{2l}\left(z_{j}+hk^{-1}-\frac{1}{2}k^{-1}+jk^{-1}+1+m\right)\right.$
$\displaystyle\left.-B_{2l}\left(hk^{-1}-\frac{1}{2}k^{-1}+jk^{-1}+1+m\right)\right).$

where $B_{2l}$ denotes the even Bernoulli polynomial (pp. 125-127 [WW21]) This
version has value at $(w,z)=\lambda^{\natural}$ exactly equal to
$f_{l}(\lambda)$. The rest of the proof is the same.

Our Theorem 2.30 now follows from this altered version of Theorem A.1 and the
altered version of $\mathrm{res}_{m,n}$.

######  Remark A.2.

Let us switch back to our notations in the second column of the table above.
In our context, when we specialize the root system to that of the
$\mathfrak{gl}$ pair, we have

$\displaystyle\rho_{\textup{SV}}$
$\displaystyle=\sum_{i=1}^{p}\left(-(p-i)-\frac{1}{2}\right)\alpha^{\textsc{B}}_{i}+\sum_{i=1}^{q}\left(-(q-i)-\frac{1}{2}+p\right)\alpha^{\textsc{F}}_{i}.$

Meanwhile, using the standard positive system, we have

$\displaystyle\rho=\sum_{i=1}^{p}(2(p-i)+1-2q)\alpha^{\textsc{B}}_{i}+\sum_{i=1}^{q}(2(q-i)+1)\alpha^{\textsc{F}}_{i}$

so

$\displaystyle\varrho=-\dfrac{1}{2}\rho=\sum_{i=1}^{p}\left(-(p-i)-\frac{1}{2}+q\right)\alpha^{\textsc{B}}_{i}+\sum_{i=1}^{q}\left(-(q-i)-\frac{1}{2}\right)\alpha^{\textsc{F}}_{i}$

and this indeed differs from $\rho_{\textup{SV}}$ in the expected way.

##  Appendix B Section 5 Computations

###  B.1. Superbracket Tables

We give the tables of superbrackets on $\mathfrak{gl}(2|2)$ with the notations
in Subsection 5.3. In the following three tables, the value of the
superbracket of the $i$-th entry in the far left column and the $j$-th entry
in the top row is given by the $(i,j)$-th entry in the table.

| $\xi_{11}$ | $\xi_{12}$ | $\xi_{21}$ | $\xi_{22}$
---|---|---|---|---
$\eta_{11}$ | $\langle 1010\rangle$ | $X_{1}$ | $Y_{2}$ | 0
$\eta_{12}$ | $X_{2}$ | 0 | $\langle 1001\rangle$ | $X_{1}$
$\eta_{21}$ | $Y_{1}$ | $\langle 0110\rangle$ | 0 | $Y_{2}$
$\eta_{22}$ | 0 | $X_{2}$ | $Y_{1}$ | $\langle 0101\rangle$

| $\xi_{11}$ | $\xi_{12}$ | $\xi_{21}$ | $\xi_{22}$
---|---|---|---|---
$X_{1}$ | $-\xi_{12}$ | 0 | $-\xi_{22}$ | 0
$X_{2}$ | 0 | 0 | $\xi_{11}$ | $\xi_{12}$
$Y_{1}$ | 0 | -$\xi_{11}$ | 0 | $-\xi_{21}$
$Y_{2}$ | $\xi_{21}$ | $\xi_{22}$ | 0 | 0

| $X_{1}$ | $X_{2}$ | $Y_{1}$ | $Y_{2}$
---|---|---|---|---
$\eta_{11}$ | 0 | $\eta_{12}$ | $-\eta_{21}$ | 0
$\eta_{12}$ | 0 | 0 | $-\eta_{22}$ | $\eta_{11}$
$\eta_{21}$ | $-\eta_{11}$ | $\eta_{22}$ | 0 | 0
$\eta_{22}$ | $-\eta_{12}$ | 0 | 0 | $\eta_{21}$

###  B.2. The Action on $\breve{W}$

Let $\\{x,h,y\\}$ be an $\mathfrak{sl}(2)$-triple with the standard relations

$[h,x]=2x,[h,y]=-2y,[x,y]=h.$

Suppose $L(m)$ is an $m+1$ dimensional irreducible $\mathfrak{sl}(2)$-module
spanned by the weight vectors

$\left\\{v_{-m},\dots,v_{-m+2i},\dots,v_{m}\right\\}$

where

$\begin{cases}h.v_{k}=kv_{k}\\\ x.v_{k}=v_{k+2}\end{cases},$

then a simple computation shows

(42) $y.v_{-m+2i}=i(m+1-i)=i(\dim V-i).$

Under the natural representation of $\mathfrak{gl}(2)$, we write

$X=\begin{pmatrix}0&1\\\ 0&0\\\ \end{pmatrix},Y=\begin{pmatrix}0&0\\\ 1&0\\\
\end{pmatrix},H=\begin{pmatrix}1&0\\\ 0&-1\\\
\end{pmatrix},Z=\begin{pmatrix}1&0\\\ 0&1\\\ \end{pmatrix},$

which give a basis for $\mathfrak{gl}(2)$ and $\\{X,H,Y\\}$ is an
$\mathfrak{sl}(2)$-triple. Denote the standard characters on the diagonal
Cartan algebra $\mathfrak{h}=\operatorname{Span}\\{H,Z\\}$ as

$\epsilon^{\pm}:\begin{pmatrix}x^{+}&0\\\ 0&x^{-}\\\ \end{pmatrix}\mapsto
x^{\pm}.$

Let $a\geq b$ be two integers. Let $L(a,b)$ be the irreducible
$\mathfrak{gl}(2)$-module of highest weight $a\epsilon^{+}+b\epsilon^{-}$ with
respect to the standard Borel subalgebra
$\operatorname{Span}\\{X,\mathfrak{h}\\}$. A weight vector basis of $L(a,b)$
can be described by

$\\{(b,a),(b+1,a-1),\dots,(a,b)\\}$

such that $h.(m,n)=(m\epsilon^{+}+n\epsilon^{-})(h)(m,n)$ for
$h\in\mathfrak{h}$ and $X.(m,n)=(m+1,n-1)$. As an $\mathfrak{sl}(2)$-module,
$L(a,b)$ is isomorphic to $L(a-b)$ by identifying $(b+i,a-i)$ with
$v_{-(a-b)+2i}$ for $i=0,1,\dots,a-b$.

For $\mathfrak{gl}(2)\oplus\mathfrak{gl}(2)$, we use subscripts $i=1,2$ for
$X,Y,H,Z$ in the $i$-th copy of $\mathfrak{gl}(2)$. The standard characters on
the second copy are denoted as $\delta^{\pm}$. Consider the irreducible
$\mathfrak{gl}(2)\oplus\mathfrak{gl}(2)$-module $L(a,b)\otimes L(c,d)$ with
the highest weight $a\epsilon^{+}+b\epsilon^{-}+c\delta^{+}+d\delta^{-}$ with
respect to the standard Borel subalgebra in both copies. Denote it as
$\breve{W}(a,b,c,d)$. Now each weight space in $\breve{W}(a,b,c,d)$ is
1-dimensional, so there is no ambiguity by setting

  1. (1)

$(b,a,d,c)$ as the lowest weight vector $(b,a)\otimes(d,c)$ (identified as
$v_{b-a}\otimes v_{d-c}$), and

  2. (2)

$X_{1}^{k}X_{2}^{l}.(b,a,d,c)=(b+k,a-k,d+l,c-l)$ (identified as
$v_{b-a+2k}\otimes v_{d-c+2l}$).

Then $Y_{1},Y_{2}$ act according to (42). In particular, we can apply this in
the proofs in Section 5.

In the proof to Theorem 5.7, the module is $\breve{W}(a,-a+2,b-1,-b-1)$. Its
first component $L(a,-a+2)$ has the lowest weight $(-a+2,a)$, corresponding to
$v_{-2a+2}\in L(2a-2)$. Note $-Y_{1}X_{1}(1,1,-1,-1)=-Y_{1}(2,0,-1,-1)$. The
weight vector $(2,0)$ corresponds to $v_{-2a+2+2a}$, and we have
$y.v_{-2a+2+2a}=a(2a-1-a)v_{0}=(a^{2}-a)v_{0}$. So
$-Y_{1}(2,0,-1,-1)=-(a^{2}-a)(1,1,-1,-1)$. Similarly
$Y_{2}X_{2}(1,1,-1,-1)=(b^{2}-b)(1,1,-1,-1)$. Together they give
$(Y_{2}X_{2}-Y_{1}X_{1})(1,1,-1,-1)=(b^{2}-a^{2}-b+a)(1,1,-1,-1)=(b+a)(b-a+1)(1,1,-1,-1)$.

In the proof to Theorem 5.8, the module is $\breve{W}(a,-a+1,a-1,-a)$. Its
first component $L(a,-a+1)$ has the lowest vector $(-a+1,1)$ corresponding to
$v_{-2a+1}\in L(2a-1)$. Note $-Y_{1}X_{1}.(1,0,-1,0)=-Y_{1}(2,-1,-1,0)$. The
weight vector $(2,-1)$ corresponds to $v_{3}=v_{-2a+1+2(a+1)}$ and
$y.v_{3}=(a+1)(2a-a-1)v_{1}$. So
$-Y_{1}.(2,-1,-1,0)=-(a+1)(2a-a-1)(1,0,-1,0)=(1-a^{2})(1,0,-1,0)$. Similarly,
$Y_{2}X_{2}(1,0,-1,0)=a^{2}(1,0,-1,0)$.

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This paper is submitted to ArXiv.org on Dec. 18, 2022.

⚽⚽⚽¡Felicidades Argentina!⚽⚽⚽


# Improving Model Compatibility of
Generative Adversarial Networks by
Boundary Calibration

Si-An Chen
National Taiwan University
<EMAIL_ADDRESS>
&Chun-Liang Li
Carnegie Mellon University
<EMAIL_ADDRESS>
Hsuan-Tien Lin
National Taiwan University
<EMAIL_ADDRESS>

###### Abstract

Generative Adversarial Networks (GANs) is a powerful family of models that
learn an underlying distribution to generate synthetic data. Many existing
studies of GANs focus on improving the realness of the generated image data
for visual applications, and few of them concern about improving the quality
of the generated data for training other classifiers—a task known as the model
compatibility problem. Literature also show that some GANs often prefer
generating ‘easier’ synthetic data that are far from the boundaries of the
classifiers, and refrain from generating near-boundary data, which are known
to play an important roles in training the classifiers. To improve GAN in
terms of model compatibility, we propose Boundary-Calibration GANs (BCGANs),
which leverage the boundary information from a set of pre-trained classifiers
using the original data. In particular, we introduce an auxiliary Boundary-
Calibration loss (BC-loss) into the generator of GAN to match the statistics
between the posterior distributions of original data and generated data with
respect to the boundaries of the pre-trained classifiers. The BC-loss is
provably unbiased and can be easily coupled with different GAN variants to
improve their model compatibility. Experimental results demonstrate that
BCGANs not only generate realistic images like original GANs but also achieves
superior model compatibility than the original GANs.

##  1 Introduction

The success of machine learning relies on not only the advances of different
models (e.g. deep learning) but also data with sufficient quality and
quantity. Nowadays, companies spend tremendous efforts and expense collecting
data to build their products. To better solve complicated real-world problems
with public or third-party machine learning experts, many companies now needs
release some data sets for competitions (e.g. Kaggle) or proof-of-concept
purposes. However, considering the costs of collecting data, companies may not
be willing to release the dataset if possible. As a result, a technique which
can generate synthetic data with properties similar to the original data is in
demand. To be specific, we are looking for generating a dataset with the
property that machine learning models trained on the generated dataset can
exhibit similar performance to ones trained on the original data. This
property is called model compatibility [24] or machine learning efficacy [32].
The organizations can share the generated data with high model compatibility
to the public and enjoy the solution derived from it without leaking the real
dataset.

When it comes to data generation, generative adversarial networks (GANs; 5) is
a popular family of generative algorithms because of its impressive
performance on generating realistic images [12]. In GANs, the generator is
trained via minimizing a neural network (discriminator) defined probability
divergence [5, 1, 22]. In addition to image generation, GANs are also widely
used in other applications, such as style transfer [11, 34, 13] and image
processing [25, 16, 2], and generating different types of data, including time
series [20, 3], text [33, 26], point clouds [18], voxels [31] and tabular data
[24, 32].

Although GANs are versatile as mentioned above, most of their development
focus on the metrics such as quality and diversity of the data [29, 9, 19].
Generating high model compatibility data via GANs is still under explored. The
pioneering work [32] first shows that data generated from conditional GANs
[21] enjoys better model compatibility than VAEs [14]. It shows a promising
potential of generating tabular data by GANs and has been used in generating
privacy-sensitive data such as clinical data [27] and insurance records [15].

In this work we wonder that whether we can further improve the model
compatibility by considering the boundary between classes, which can be
approximated by the models trained on the original data. For example,
Wasserstein GAN (WGAN; 1) performs a mean-matching between the distribution of
real data and generated data. However, only mean-matching is sometimes not
enough to learn the whole distribution especially for those boundary cases.
Apparently, if a GAN knows the boundary between different classes, it may be
able to generate instances which are close to the boundary with correct
labels. These boundary points will guide a classifier to learn the correct
decision boundary.

In this work, we try to improve GANs with regards to model compatibility in
classification problems. We use a set of pre-trained classifiers to obtain
multiple decision boundaries. Then use an auxiliary loss function called
Boundary-Calibration loss (BC-loss) to calibrate the generating distribution
according to the decision boundaries of these pre-trained classifiers. The
main contributions of this work are:

  * •

In Section 2, we propose a way to evaluate model compatibility in
classification problems. We consider a variety of machine learning algorithms
and average the performance to obtain a comprehensive metric.

  * •

In Section 4, we propose a loss function called Boundary-Calibration loss (BC-
loss) which helps typical GANs to learn a distribution with better model
compatibility. The loss considers the decision boundaries of pre-trained
classifiers and minimizes the maximum mean discrepancy (MMD; 6) between the
original dataset and the generated dataset. In addition, we show that
optimizing the BC-loss would not change the optimal solution of the original
GAN, but it reduces the feasible set to ensure the model compatibility.

  * •

In Section 5, we demonstrate how BC-loss affects the boundary of the generated
data with a two-dimensional toy dataset. We also show that the BC-loss
improves model compatibility of the generated data with different types of
classifiers and a variety of datasets. Finally, we inspect the feature
selection results to examine how the interpretation of machine learning models
may be effected.

Last, in Section 3, we discuss some works which are similar to our work and
describe how does our work differ from them.

##  2 Model compatibility in classification

In this work, we focus on generating data for fully-supervised classification
learning. Given a dataset $D=\\{({\bm{x}}_{i},y_{i})\\}_{i=1}^{n}$, where
${\bm{x}}_{i}\in\mathcal{X}$ represents features of an instance,
$y_{i}=f({\bm{x}}_{i})\in\mathcal{Y}$ represents the corresponding label of
${\bm{x}}_{i}$ according $f:\mathcal{X}\rightarrow\mathcal{Y}$, and
$({\bm{x}}_{i},y_{i})\sim P_{D}$, a learning algorithm
$A:(\mathcal{X},\mathcal{Y})^{n}\rightarrow\mathcal{H}$ learns a hypothesis
$h\in\mathcal{H}$ to approximate the mapping function, i.e. $A(D)=h\approx f$.
Our goal is to obtain a generator $G$ which generates a synthetic dataset
$D^{\prime}=\\{({\bm{x}}_{j}^{\prime},y_{j}^{\prime})\\}_{j=1}^{m}$ such that
$A(D^{\prime})=h^{\prime}\approx h$. We call this property model compatibility
as proposed in Park et al. 24.

To measure the model compatibility of a generated dataset quantitatively, we
consider the performance of a classifier trained on the generated dataset
comparing to the one trained on the real dataset. We evaluate the accuracy on
a separate test dataset to indicate the performance of a given classifier. In
addition, we calculate relative accuracy by scaling the test accuracy of the
classifier trained on the generated dataset by the accuracy of the classifier
trained on the real dataset. The relative accuracy allows us to average the
results from different machine learning algorithms more fairly. The final
evaluation is :

$\frac{1}{|{\mathbb{A}}|}\sum_{A\in{\mathbb{A}}}\frac{acc(h^{\prime},D^{(t)})}{acc(h,D^{(t)})},$
(1)

where ${\mathbb{A}}$ is a set of learning algorithms ,
$D^{(t)}=\\{(x_{i}^{(t)},y_{i}^{(t)})\\}_{i=1}^{N}$ is the test dataset, and
$acc(h,D^{(t)})$ is the accuracy of hypothesis $h$ on test data $D^{(t)}$. We
can determine ${\mathbb{A}}$ as a wide variety of learning algorithms to make
the metric provide a more comprehensive measurement of model compatibility.

##  3 Related work

Research about generating data for classification can be divided into two
categories: formulation and architecture. For formulation, Conditional GAN
(CGAN; 21 ) is an intuitive way to generate instances with corresponding
labels. We can learn the distribution of labels by counting and sample the
instances from CGAN conditionally. Auxiliary Classifier GAN (ACGAN; 23) is
considered as a better way for conditional generation. It uses an auxiliary
classifier to provide information about the boundary between each classes.
However, ACGAN has been proved that the objective is biased so it tends to
generate data with lower entropy for the auxiliary classifier [30]. Thus, the
lose of instances near the decision boundary may worsen the model
compatibility. In this work, we use CGAN along with the proposed BC-loss to
generate data with model compatibility.

On the other hand, the other line of research focuses on generating data with
different network architecture or data processing procedure. Recent works that
also consider model compatibility are Table GAN [24] and Tabular GAN [32].
Table GAN focuses on the privacy of generated data and thus their is a trade
off between privacy and model compatibility. To achieve privacy preserving,
they do not improve the model compatibility compared to the original GAN. On
the other hand, Tabular GAN puts emphasis on increasing model compatibility of
generated data. They propose a framework with a more complicated data
processing procedure and use LSTM to better parameterize the target
distribution. In contrast to these works, our work focus on the formulation of
GANs and can be applied to most variants of GANs, including Table-GAN and
Tabular GAN. Moreover, while these former works only focus on tabular data,
our BC-loss is applicable to generate image datasets as well.

Some GAN variants are named similarly to our work but they pay attention to
different problems. For example, the boundary described in boundary-seeking
GAN [10] means the decision boundary of the discriminator rather than the
decision boundary for the supervised labels. To the best of our knowledge, we
are the first work trying to improve model compatibility by modifying the
formulation of GANs.

##  4 Boundary-Calibration GAN

To achieve better model compatibility of GAN, we propose an auxiliary GAN loss
which we call boundary-calibration loss (or BC-loss). We assume that we have a
set of pre-trained classifiers which are well-trained on the original dataset.
The BC-loss helps GANs to calibrate the distribution with respect to the
distribution of decision values predicted by pre-trained classifiers. The
calibration leads to more accurate data generation near the decision boundary
and thus enabling a machine learning algorithm to learn a similar hypothesis
to one that learns from the original dataset. To ease the complexity of
learning to generate $({\bm{x}},y)$ jointly, we infer $P(y)$ by counting the
proportion of each class in the original dataset and train a conditional
generator $G$ such that $G({\bm{z}},y)\sim P_{\mathcal{X}|y}$, where
$P_{\mathcal{X}|y}$ is the conditional data distribution and ${\bm{z}}\sim
P_{\mathcal{Z}}$ is the initial randomness such as Gaussian distribution.
Therefore we can generate $({\bm{x}},y)$ by sampling $y\sim P(y)$ and
$G({\bm{z}},y)$.

###  4.1 Boundary Calibration

Given a pre-trained classifier $C$, we hope the generated dataset will adopt
the same statistics as the original dataset while considering the decision
boundary of $C$. To include the information about the boundary, we calculate
posterior $P_{C}(y\mid{\bm{x}}_{i})$ from the decision values predicted by the
classifier. In practice, we can apply a softmax function to the outputs of a
classifier to obtain the posterior. The posterior provides information of an
instance from the classifier’s aspect. Therefore, given the real dataset
$X=\\{{\bm{x}}_{1},{\bm{x}}_{2},...,{\bm{x}}_{n}\\}$, we can obtain a set of
posterior vector
$C(X)=(P_{C}(y\mid{\bm{x}}_{1}),P_{C}(y\mid{\bm{x}}_{2}),...,P_{C}(y\mid{\bm{x}}_{n}))$.
To generate data $X^{\prime}$ with the same distribution of posteriors to the
boundary, we match the statistics of $C(X)$ and $C(X^{\prime})$ by optimizing
a distance $M$:

$\mathcal{L}_{BC}(X,X^{\prime},C)=M(C(X),C(X^{\prime}))$ (2)

Here $M$ can be any distance metric which measures the distance between two
sets of samples. In statistics, distinguishing whether two sets of samples are
from the same distribution is called Two-Sample Test. A classical solution to
two-sample test is kernel maximum mean discrepancy (MMD; 6). The idea is to
compare the statistics between the two sets of samples. If the statistics are
similar then these two sets might be sampled from the same distribution. Given
two sets of samples $X=\\{{\bm{x}}_{i}\\}_{i=1}^{n}$ and
$Y=\\{{\bm{y}}_{j}\\}_{j=1}^{n}$, an unbiased estimator of MMD with kernel $k$
is defined as:

$\hat{M}_{k}(X,Y)=\frac{1}{\binom{n}{2}}\sum_{i\neq
i^{\prime}}k({\bm{x}}_{i},{\bm{x}}_{i^{\prime}})-\frac{2}{\binom{n}{2}}\sum_{i\neq
j}k({\bm{x}}_{i},{\bm{y}}_{j})+\frac{1}{\binom{n}{2}}\sum_{j\neq
j^{\prime}}k({\bm{y}}_{j},{\bm{y}}_{j^{\prime}})$ (3)

In practice, we use Gaussian kernel
$k({\bm{x}},{\bm{x}}^{\prime})=\exp(\|{\bm{x}}-{\bm{x}}^{\prime}\|^{2})$ in
MMD since Gaussian kernel is a characteristic kernel which ensures that the
distance is zero if and only if the two distributions are the same [6].

The BC-loss can be applied in generator of any GAN variants to improve the
model compatibility. In addition, to better fit the real unknown boundary, we
can use multiple classifiers to calibrate the distributions from different
aspects. As a result, for a loss function of generator $\mathcal{L}_{G}$, we
can modify the loss to be:

$\hat{\mathcal{L}}_{G}=\mathcal{L}_{G}+\frac{\lambda}{|{\mathbb{C}}|}\sum_{C\in{\mathbb{C}}}\hat{M}_{k}(C(X),C(G({\mathbb{Z}},\mathcal{Y})))$
(4)

where ${\mathbb{Z}}$ is a set of noises, ${\mathbb{C}}$ is a set of pre-
trained classifiers and $\lambda$ is a hyper-parameter to control the weight
of BC-loss.

###  4.2 Analysis of optimal solution

Next we prove that adding our proposed BC-loss would not change the optimal
solution of the original objective. Here we assume the loss of the generator
$\mathcal{L}_{G}$ achieves its optimal value in the its GAN objectives
$\mathcal{L}_{G}$ if and only if the distribution of $G({\bm{z}},y)$ recovers
$P_{\mathcal{X}|y}$ for all $y\in\mathcal{Y}$, which holds for the vanilla GAN
[5] and most of other GAN variants.

######  Theorem 1 (Gretton et al. 6).

Given a kernel $k$, if $k$ is a characteristic kernel, then $M_{k}(P,Q)=0\iff
P=Q$.

######  Theorem 2 (Equivalence of optimal solution).

$G$ is an optimal solution of $\mathcal{L}_{G}$ $\iff$ $G$ is an optimal
solution of $\hat{\mathcal{L}}_{G}$

###### Proof.

$(\Rightarrow)$ According to the assumption, $G$ is an optimal solution of
$\mathcal{L}_{G}$ implies $G(\mathcal{Z},y)$ recovers $P_{\mathcal{X}|y}$ for
all $y\in\mathcal{Y}$. Therefore,
$P_{C(X)}=P_{C(G({\mathbb{Z}},\mathcal{Y}))}$ and
$M_{k}(P_{C(X)},P_{C(G({\mathbb{Z}},\mathcal{Y}))})=0$ by Theorem 1. Now
$\hat{\mathcal{L}}_{G}=\mathcal{L}_{G}+0=\mathcal{L}_{G}$ and $G$ is an
optimal solution of $\mathcal{L}_{G}$, so $G$ is also an optimal solution of
$\hat{\mathcal{L}}_{G}$.

$(\Leftarrow)$ Since $\mathcal{L}_{BC}\geq 0$, we have
$\hat{\mathcal{L}}_{G}=\mathcal{L}_{G}+\mathcal{L}_{BC}\geq\mathcal{L}_{G}$.
From above, we know $\hat{\mathcal{L}}_{G}(G)=0$ if $G=P_{\mathcal{X}}$. Thus,
for an optimal solution $G^{*}$,
$0\geq\hat{\mathcal{L}}_{G}(G^{*})\geq\mathcal{L}_{G}(G^{*})\geq 0$, which
implies $\hat{\mathcal{L}}_{G}(G^{*})=\mathcal{L}_{G}(G^{*})=0$. Therefore,
$G^{*}$ is also an optimal solution of $\mathcal{L}_{G}$. ∎

The proof shows that the proposed BC-loss does not change the optimal solution
of the original optimization problem. However, we can consider BC-loss as a
Lagrangian constraint which restricts the solution to a subspace where the
generator owns higher model compatibility .

###  4.3 Comparison to MMD GAN

MMD GAN [17] is a variant of GAN where the generator tries to minimize the MMD
between generated data and original data and the discriminator learns a kernel
which maximizes the MMD. Though the formulation of MMD GAN and BC-loss are
similar, they still do not conflict because MMD GAN do not known the
information about the classifier and the objective of MMD GAN would not lead
the discriminator to a classifier. Therefore, BC-loss may still improve MMD
GAN by guiding the generator to not generate points across the boundary. To
understand the improvement in MMD GAN from BC-loss, we use MMD GAN as one of
the baselines in our experiments.

##  5 Experiments

In this section, we use a toy dataset to illustrate how the proposed method
improves the model compatibility. To be more realistic, we provide more
comprehensive results for four different real-world dataset from UCI dataset
repository [4]: Adult, Connect-4, Covertype and Sensorless. We then show our
method is also applicable in image dataset: MNIST and Cifar10 without losing
the image quality. In addition, we investigate the results of feature
selections on the generated dataset to see whether the generated data can
preserve the interpretation of machine learning models.

###  5.1 Experimental settings

#### Evaluation

In this work, we focus on model compatibility of generated datasets. We use a
wide variety of machine learning algorithms including linear SVM, decision
tree (DT), random forest (RF), and multi-layer perception (MLP) to evaluate
the model compatibility. As described in Section 2, we evaluate the relative
accuracy for each type of machine learning model, where the relative accuracy
is calculated by dividing the accuracy of classifier trained on generated data
to the accuracy of classifier trained on original data.

#### Compared methods

We take Wasserstein GAN (WGAN) and MMD GAN as our baselines to evaluate the
effectiveness of the proposed boundary-calibration technique. We denotes their
counterparts with BC-loss as BWGAN and BMMDGAN respectively. All of the
methods use gradient penalty to enforce the Lipschitz constraint on the
discriminator [7, 17]. To achieve conditional data generation as described in
Section 4, we add an embedding layer to learn the embedding vector for each
class. The embedding vector is concatenated as additional input features for
both generators and discriminators on UCI datasets. For image datasets, the
embedding vectors are used as described in [21].

###  5.2 2D Toy Dataset

(a) Real
accuracy = 0.995

(b) ACGAN
accuracy = 0.932

(c) WGAN
accuracy = 0.979

(d) BWGAN (proposed)
accuracy = 0.984

Figure 1: A toy dataset generated by different GAN methods. Figure (a) is the
original training data and the others are data generated by ACGAN, WGAN and
our BWGAN respectively. The background color indicates the decision boundary
of a random forest trained on corresponding data. The captions show the test
accuracy of the random forest.

We use a 2D toy dataset with two classes to illustrate the results generated
by different GAN methods in Figure 1. Figure 1(a) shows the distribution of
the original training data. We use these generated data to train a random
forest and depict the decision boundary by different background color. From
Figure 1(b), we can see that although ACGAN can make use of the auxiliary
classifier during training, it learns a biased distribution that push the
generated data away from the boundary. The large margin between the two
clusters brings more uncertainty to the decision boundary and thus leads to
worse test accuracy. In Figure 1(c), WGAN approximates the original
distribution well in the center part of the two cluster, but do not get a
clear boundary between the two classes. It generates some ambiguous points
near the boundary that would confuse the classifier. Finally, our BWGAN
generates points near the boundary more precisely, as shown in Figure 1(c).

###  5.3 UCI dataset

We evaluate our proposed BC-GAN on four datasets from UCI repository. The
attributes of the datasets can be found in Appendix A. Discrete features are
processed to one-hot encoding and continuous features are scaled to $[0,1]$.
For each dataset, we train six multi-layer perceptrons with a random split of
half of training data as pre-trained classifiers. In these experiments,
generators and discriminators are consist of 3 fully-connected hidden layer
with 128 units. A logistic function is applied to the output layer of
generators to generate features within 0 and 1. The weight of BC-loss is set
to be $\lambda=100$ for all datasets.

Table 1 summarize the comparison between different methods. We calculate the
relative accuracy of different machine learning models mentioned in Section
5.1 and average the relative accuracy to indicate the model compatibility of
generated data for each dataset. The table shows that the proposed BC-loss
improves the accuracy of classifiers generally compared to original WGAN and
MMD GAN. Moreover, ACGAN performs worst on three out of four datasets and
exhibit a significant deficiency though it is proved to have the state-of-the-
art generation quality. This again prove that the biased objective of ACGAN
worsen the model compatibility seriously.The breakdown results and real
accuracy are provided in Appendix B.

To further investigate the advantage of boundary-calibration, we visualize the
generated results of Sensorless in Figure 2. We train a fully-connected neural
network with a 2-units hidden layer before the output layer to project the
generated samples to a 2-dimensional embedding space. The projection
classifier is well-trained and achieves over 99% testing accuracy so we can
use it to determine whether a sample is generated with incorrect label. The
figure shows that there are less mislabeled data generated by BWGAN,
especially at the center and the bottom-left region. The fact indicates that
boundary-calibration helps GANs generate labeled data more accurately, which
may lead to the improvement of classification accuracy.

Table 1: Summary result of model compatibility evaluate on UCI datasets. The numbers are relative accuracy. | adult | connect4 | covertype | sensorless | average
---|---|---|---|---|---
ACGAN | 97.78 | 83.71 | 51.98 | 77.47 | 77.74
WGAN | 96.60 | 87.59 | 79.56 | 84.63 | 87.10
BWGAN | 98.79 | 88.95 | 83.16 | 93.34 | 91.06
MMDGAN | 95.67 | 86.29 | 77.14 | 86.28 | 86.35
BMMDGAN | 97.23 | 87.38 | 79.82 | 88.14 | 88.14

(a) Real
mislabeled = 0.8%

(b) ACGAN
mislabeled = 12.57%

(c) WGAN
mislabeled = 9.35%

(d) BWGAN (proposed)
mislabeled=7.39%

Figure 2: 2D visualization of real and generated Sensorless dataset. The
mislabeled points are emphasized with border lines. The background color
indicates the spaces of each class according to the projection classifier.

###  5.4 Interpretability

In addition to accuracy, it is also important that the model trained on
generated data should give us the same interpretation of a model trained on
the original data. We investigate the interpretability by two common feature
selection techniques. First, we train two random forests on the generated and
original dataset respectively. Each random forest can provide the importances
of the features. We evaluate the consistency of interpretation by calculating
precision at Kth, which means how many features ranked top-k in random forest
trained on original data are in the top-k importance feature of the random
forest trained on generated data. The results are shown in Table 2. We provide
the results of training a classifier on the same original data with a
different random seed as REAL for comparison. The effect of BC-loss is not
significant in this aspect. However, the scores of ACGAN drop seriously, which
means training a classifier on data generated by ACGAN is somehow dangerous
because the meaning of model may be totally different.

Another way to select feature is training a linear model with $\ell_{1}$
regularization. In Table 3 we use linear SVM with $\ell_{1}$ regularization to
select features. Then we calculate the F1 score of features selected by
classifiers trained on generated data to known how similar between the two
sets of features selected by classifiers trained on original and generated
dataset. The results again shows that using boundary-calibration does not has
significant effect to feature selection and ACGAN is not proper to generated
data for training.

Table 2: Precision at K of feature importance ranking compared to the feature importance ranking obtained from the original dataset |  | REAL | ACGAN | WGAN | BWGAN | MMDGAN | BMMDGAN
---|---|---|---|---|---|---|---
dataset | metric |  |  |  |  |  |
adult | P@10 | 0.80 | 0.40 | 0.80 | 0.80 | 0.80 | 0.70
| P@20 | 0.95 | 0.45 | 0.75 | 0.80 | 0.85 | 0.75
| P@30 | 1.00 | 0.63 | 0.90 | 0.90 | 0.87 | 0.87
connect4 | P@10 | 0.90 | 0.30 | 1.00 | 1.00 | 0.90 | 0.80
| P@20 | 0.90 | 0.35 | 0.90 | 0.85 | 0.95 | 0.85
| P@30 | 0.93 | 0.33 | 0.83 | 0.90 | 0.77 | 0.83
covertype | P@10 | 0.80 | 0.30 | 0.60 | 0.80 | 0.30 | 0.60
| P@20 | 1.00 | 0.50 | 0.75 | 0.85 | 0.90 | 0.85
| P@30 | 1.00 | 0.70 | 0.93 | 0.87 | 0.87 | 0.87
sensorless | P@10 | 0.80 | 0.60 | 0.80 | 0.90 | 0.70 | 0.70
| P@20 | 0.95 | 0.50 | 0.75 | 0.85 | 0.80 | 0.85
| P@30 | 1.00 | 0.80 | 0.90 | 0.90 | 0.87 | 0.83
Table 3: F1 score of feature selection by $\ell_{1}$-regularized linear SVM |  | REAL | ACGAN | WGAN | BWGAN | MMDGAN | BMMDGAN
---|---|---|---|---|---|---|---
dataset | metric |  |  |  |  |  |
adult | f1 (C=0.01) | 0.975 | 0.571 | 0.697 | 0.787 | 0.795 | 0.725
| f1 (C=0.001) | 0.968 | 0.500 | 0.737 | 0.789 | 0.700 | 0.789
connect4 | f1 (C=0.01) | 0.905 | 0.860 | 0.889 | 0.866 | 0.874 | 0.831
| f1 (C=0.001) | 0.933 | 0.718 | 0.796 | 0.739 | 0.750 | 0.752
covertype | f1 (C=0.01) | 0.989 | 0.923 | 0.911 | 0.935 | 0.730 | 0.773
| f1 (C=0.001) | 1.000 | 0.825 | 0.912 | 0.825 | 0.800 | 0.815
sensorless | f1 (C=0.01) | 0.982 | 0.848 | 0.900 | 0.918 | 0.813 | 0.844
| f1 (C=0.001) | 0.815 | 0.778 | 0.769 | 0.733 | 0.812 | 0.710

###  5.5 Image dataset

(a) WGAN MNIST

(b) BWGAN MNIST

(c) WGAN CIFAR-10 (Inception: 6.00, FID: 49.47)

(d) BWGAN CIFAR-10 (Inception: 6.12, FID: 44.93)

Figure 3: Gnerated samples from WGAN and our BWGAN. The images in the same
column are in the same category.

We further use MNIST and CIFAR-10 dataset to investigate the effectiveness of
boundary-calibration on image datasets. For MNIST, we train six 4-layer
convolution neural networks (CNN) with random sampling half of training data
as pre-trained classifiers, and use the same classifier set in Section 5.1 to
evaluate model compatibility. For CIFAR-10, we use ResNet56v2 [8] to obtain
three pre-trained classifier and evaluate on CNN and ResNet56v2. In both task,
we use DCGAN [28] as network structure in all GANs. The weight of BC-loss is
set to be $\lambda=1$ for these two datasets.

Table 5 and Table 5 show the relative accuracy of classifiers trained on
generated data. The proposed BWGAN still outperforms WGAN with better accuracy
in general. The results generated by WGAN and BWGAN are pictured in Figure 3.
The Inception score and Frechet Inception Distance (FID) for CIFAR-10 are also
provided in the caption of Figure 3. Though the difference of quality between
the images generated from WGAN and BWGAN is not significant in visual, the
quantitative scores for quality of generated samples of CIFAR-10 are slightly
improved. The results indicate that even though our method seems not improve
the image quality, it is still able to improve the model compatibility without
losing image quality.

Table 4: Breakdown results on MNIST dataset. | REAL | WGAN | BWGAN
---|---|---|---
DT (d=10) | 86.6 | 54.5 (63.0) | 48.3 (55.8))
DT (d=20) | 88.0 | 34.5 (39.2) | 48.2 (54.8)
Linear SVM | 88.0 | 34.5 (39.2) | 48.2 (54.8)
MLP (100) | 97.6 | 96.2 (98.5) | 96.8 (99.2)
MLP (200x2) | 97.9 | 96.7 (98.8) | 96.1 (98.2)
RF (n=10, d=10) | 92.5 | 75.5 (81.7) | 83.7 (90.6)
RF (n=10, d=20) | 94.7 | 67.5 (71.3) | 71.5 (75.4)
Avg. | 100.0 | 70.2 | 75.5

Table 5: Breakdown results on CIFAR-10 dataset. | REAL | WGAN | BWGAN
---|---|---|---
CNN | 70.8 | 63.5 (89.8) | 63.0 (89.1)
Resnet56v2 | 77.5 | 48.8 (62.9) | 51.3 (66.2)
Avg. | 100.0 | 76.4 | 77.6

##  6 Discussion

We introduce an auxiliary loss in GANs which improves the model compatibility
of generated dataset. We prove the new loss is unbiased and is applicable to
all variants of GAN to improve model compatibility. We further demonstrate
that our method has clear advantages with a variety of machine learning models
trained on generated dataset. In addition, we investigate the results of
feature selection and found that the BC-loss doesn’t effect the interpretation
of machine learning models. While this work only focus on classification
problem, generating data for regression problem is also worth studying. We
hope our work will open the path for GANs with better model compatibility so
that synthetic data can be more useful in practice.

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##  Appendix A Dataset Information

Table 6: Attributes of UCI datasets Dataset | # of train | # of test | # of discrete feature | # of continuous feature | # of class
---|---|---|---|---|---
Adult | 32561 | 16281 | 123 | 0 | 2
Connect-4 | 54046 | 13511 | 126 | 0 | 3
Covertype | 116203 | 116202 | 44 | 10 | 7
Sensorless | 46807 | 11702 | 0 | 48 | 11

##  Appendix B Detail Result

###  B.1 Adult

| REAL | ACGAN | WGAN | BWGAN | MMDGAN | BMMDGAN
---|---|---|---|---|---|---
DT (d=10) | 83.6 (100.0) | 80.8 (96.6) | 80.3 (96.0) | 81.9 (97.9) | 79.7 (95.3) | 81.2 (97.1)
DT (d=20) | 81.2 (100.0) | 80.9 (99.6) | 75.9 (93.5) | 79.9 (98.4) | 73.8 (91.0) | 75.9 (93.5)
Linear SVM | 81.2 (100.0) | 80.9 (99.6) | 75.9 (93.5) | 79.9 (98.4) | 73.8 (91.0) | 75.9 (93.5)
MLP (100) | 84.4 (100.0) | 81.8 (96.9) | 83.1 (98.5) | 83.6 (99.1) | 82.9 (98.2) | 84.1 (99.7)
MLP (200x2) | 84.4 (100.0) | 82.0 (97.2) | 83.1 (98.5) | 83.8 (99.3) | 82.6 (97.9) | 84.2 (99.8)
RF (n=10, d=10) | 83.9 (100.0) | 82.6 (98.4) | 83.4 (99.4) | 83.6 (99.6) | 83.5 (99.5) | 83.3 (99.3)
RF (n=10, d=20) | 84.1 (100.0) | 80.7 (96.0) | 81.5 (96.9) | 83.2 (99.0) | 81.5 (96.9) | 82.3 (97.8)
Avg. | 100.0 | 97.8 | 96.6 | 98.8 | 95.7 | 97.2

###  B.2 Connect4

| REAL | ACGAN | WGAN | BWGAN | MMDGAN | BMMDGAN
---|---|---|---|---|---|---
DT (d=10) | 74.7 (100.0) | 64.1 (85.8) | 67.7 (90.7) | 69.5 (93.1) | 66.7 (89.3) | 68.5 (91.7)
DT (d=20) | 76.3 (100.0) | 64.1 (83.9) | 60.9 (79.8) | 64.9 (85.1) | 61.4 (80.4) | 64.1 (84.0)
Linear SVM | 76.3 (100.0) | 64.1 (83.9) | 60.9 (79.8) | 64.9 (85.1) | 61.4 (80.4) | 64.1 (84.0)
MLP (100) | 84.2 (100.0) | 66.0 (78.4) | 74.7 (88.7) | 73.7 (87.6) | 72.2 (85.8) | 72.8 (86.5)
MLP (200x2) | 85.7 (100.0) | 66.6 (77.7) | 74.5 (86.9) | 73.7 (86.0) | 71.9 (83.8) | 72.9 (85.0)
RF (n=10, d=10) | 73.0 (100.0) | 66.8 (91.6) | 71.4 (97.8) | 70.4 (96.4) | 70.1 (96.0) | 68.3 (93.6)
RF (n=10, d=20) | 79.2 (100.0) | 67.0 (84.6) | 70.7 (89.3) | 70.8 (89.4) | 69.8 (88.1) | 68.9 (87.0)
Avg. | 100.0 | 83.7 | 87.6 | 88.9 | 86.3 | 87.4

###  B.3 Covertype

| REAL | ACGAN | WGAN | BWGAN | MMDGAN | BMMDGAN
---|---|---|---|---|---|---
DT (d=10) | 77.1 (100.0) | 37.8 (49.1) | 65.6 (85.1) | 68.9 (89.4) | 65.5 (84.9) | 66.7 (86.5)
DT (d=20) | 86.9 (100.0) | 39.4 (45.3) | 62.9 (72.3) | 67.2 (77.3) | 62.2 (71.6) | 63.4 (73.0)
Linear SVM | 86.9 (100.0) | 39.4 (45.3) | 62.9 (72.3) | 67.2 (77.3) | 62.2 (71.6) | 63.4 (73.0)
MLP (100) | 80.6 (100.0) | 54.0 (67.0) | 67.5 (83.8) | 69.9 (86.7) | 62.5 (77.5) | 66.2 (82.2)
MLP (200x2) | 89.2 (100.0) | 53.6 (60.1) | 66.5 (74.5) | 68.3 (76.5) | 58.8 (65.9) | 65.1 (73.0)
RF (n=10, d=10) | 73.8 (100.0) | 38.6 (52.2) | 67.2 (91.0) | 69.5 (94.1) | 68.0 (92.1) | 67.8 (91.8)
RF (n=10, d=20) | 86.2 (100.0) | 38.6 (44.8) | 67.2 (77.9) | 69.7 (80.8) | 65.9 (76.4) | 68.3 (79.2)
Avg. | 100.0 | 52.0 | 79.6 | 83.2 | 77.1 | 79.8

###  B.4 Sensorless

| REAL | ACGAN | WGAN | BWGAN | MMDGAN | BMMDGAN
---|---|---|---|---|---|---
DT (d=10) | 96.3 (100.0) | 75.8 (78.7) | 77.9 (80.8) | 87.2 (90.5) | 82.3 (85.5) | 82.6 (85.8)
DT (d=20) | 98.4 (100.0) | 75.4 (76.7) | 68.3 (69.5) | 89.6 (91.1) | 81.3 (82.6) | 83.8 (85.2)
Linear SVM | 98.4 (100.0) | 75.4 (76.7) | 68.3 (69.5) | 89.6 (91.1) | 81.3 (82.6) | 83.8 (85.2)
MLP (100) | 93.6 (100.0) | 72.8 (77.8) | 87.8 (93.8) | 89.8 (96.0) | 82.4 (88.0) | 84.9 (90.6)
MLP (200x2) | 98.7 (100.0) | 76.8 (77.8) | 90.5 (91.7) | 93.7 (94.9) | 85.4 (86.5) | 87.7 (88.8)
RF (n=10, d=10) | 98.4 (100.0) | 76.2 (77.5) | 92.1 (93.6) | 92.5 (94.1) | 87.9 (89.4) | 88.7 (90.2)
RF (n=10, d=20) | 99.8 (100.0) | 77.1 (77.3) | 93.3 (93.6) | 95.6 (95.8) | 89.2 (89.4) | 90.8 (91.0)
Avg. | 100.0 | 77.5 | 84.6 | 93.3 | 86.3 | 88.1